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In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence {G_n = (V_n, E_n)} of finite graphs, one can extract a subsequence {G_{n_j} = (V_{n_j}, E_{n_j})} which converges (in a specific sense) to a continuous object known as a “graphon” – a symmetric measurable function {p\colon [0,1] \times [0,1] \rightarrow [0,1]}. What “converges” means in this context is that subgraph densities converge to the associated integrals of the graphon {p}. For instance, the edge density

\displaystyle  \frac{1}{|V_{n_j}|^2} |E_{n_j}|

converge to the integral

\displaystyle  \int_0^1 \int_0^1 p(x,y)\ dx dy,

the triangle density

\displaystyle  \frac{1}{|V_{n_j}|^3} \lvert \{ (v_1,v_2,v_3) \in V_{n_j}^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_{n_j} \} \rvert

converges to the integral

\displaystyle  \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ dx_1 dx_2 dx_3,

the four-cycle density

\displaystyle  \frac{1}{|V_{n_j}|^4} \lvert \{ (v_1,v_2,v_3,v_4) \in V_{n_j}^4: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_4\}, \{v_4,v_1\} \in E_{n_j} \} \rvert

converges to the integral

\displaystyle  \int_0^1 \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_4) p(x_4,x_1)\ dx_1 dx_2 dx_3 dx_4,

and so forth. One can use graph limits to prove many results in graph theory that were traditionally proven using the regularity lemma, such as the triangle removal lemma, and can also reduce many asymptotic graph theory problems to continuous problems involving multilinear integrals (although the latter problems are not necessarily easy to solve!). See this text of Lovasz for a detailed study of graph limits and their applications.

One can also express graph limits (and more generally hypergraph limits) in the language of nonstandard analysis (or of ultraproducts); see for instance this paper of Elek and Szegedy, Section 6 of this previous blog post, or this paper of Towsner. (In this post we assume some familiarity with nonstandard analysis, as reviewed for instance in the previous blog post.) Here, one starts as before with a sequence {G_n = (V_n,E_n)} of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter {\alpha \in\beta {\bf N} \backslash {\bf N}}) to obtain a nonstandard graph {G_\alpha = (V_\alpha,E_\alpha)}, where {V_\alpha = \prod_{n\rightarrow \alpha} V_n} is the ultraproduct of the {V_n}, and similarly for the {E_\alpha}. The set {E_\alpha} can then be viewed as a symmetric subset of {V_\alpha \times V_\alpha} which is measurable with respect to the Loeb {\sigma}-algebra {{\mathcal L}_{V_\alpha \times V_\alpha}} of the product {V_\alpha \times V_\alpha} (see this previous blog post for the construction of Loeb measure). A crucial point is that this {\sigma}-algebra is larger than the product {{\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha}} of the Loeb {\sigma}-algebra of the individual vertex set {V_\alpha}. This leads to a decomposition

\displaystyle  1_{E_\alpha} = p + e

where the “graphon” {p} is the orthogonal projection of {1_{E_\alpha}} onto {L^2( {\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha} )}, and the “regular error” {e} is orthogonal to all product sets {A \times B} for {A, B \in {\mathcal L}_{V_\alpha}}. The graphon {p\colon V_\alpha \times V_\alpha \rightarrow [0,1]} then captures the statistics of the nonstandard graph {G_\alpha}, in exact analogy with the more traditional graph limits: for instance, the edge density

\displaystyle  \hbox{st} \frac{1}{|V_\alpha|^2} |E_\alpha|

(or equivalently, the limit of the {\frac{1}{|V_n|^2} |E_n|} along the ultrafilter {\alpha}) is equal to the integral

\displaystyle  \int_{V_\alpha} \int_{V_\alpha} p(x,y)\ d\mu_{V_\alpha}(x) d\mu_{V_\alpha}(y)

where {d\mu_V} denotes Loeb measure on a nonstandard finite set {V}; the triangle density

\displaystyle  \hbox{st} \frac{1}{|V_\alpha|^3} \lvert \{ (v_1,v_2,v_3) \in V_\alpha^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_\alpha \} \rvert

(or equivalently, the limit along {\alpha} of the triangle densities of {E_n}) is equal to the integral

\displaystyle  \int_{V_\alpha} \int_{V_\alpha} \int_{V_\alpha} p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ d\mu_{V_\alpha}(x_1) d\mu_{V_\alpha}(x_2) d\mu_{V_\alpha}(x_3),

and so forth. Note that with this construction, the graphon {p} is living on the Cartesian square of an abstract probability space {V_\alpha}, which is likely to be inseparable; but it is possible to cut down the Loeb {\sigma}-algebra on {V_\alpha} to minimal countable {\sigma}-algebra for which {p} remains measurable (up to null sets), and then one can identify {V_\alpha} with {[0,1]}, bringing this construction of a graphon in line with the traditional notion of a graphon. (See Remark 5 of this previous blog post for more discussion of this point.)

Additive combinatorics, which studies things like the additive structure of finite subsets {A} of an abelian group {G = (G,+)}, has many analogies and connections with asymptotic graph theory; in particular, there is the arithmetic regularity lemma of Green which is analogous to the graph regularity lemma of Szemerédi. (There is also a higher order arithmetic regularity lemma analogous to hypergraph regularity lemmas, but this is not the focus of the discussion here.) Given this, it is natural to suspect that there is a theory of “additive limits” for large additive sets of bounded doubling, analogous to the theory of graph limits for large dense graphs. The purpose of this post is to record a candidate for such an additive limit. This limit can be used as a substitute for the arithmetic regularity lemma in certain results in additive combinatorics, at least if one is willing to settle for qualitative results rather than quantitative ones; I give a few examples of this below the fold.

It seems that to allow for the most flexible and powerful manifestation of this theory, it is convenient to use the nonstandard formulation (among other things, it allows for full use of the transfer principle, whereas a more traditional limit formulation would only allow for a transfer of those quantities continuous with respect to the notion of convergence). Here, the analogue of a nonstandard graph is an ultra approximate group {A_\alpha} in a nonstandard group {G_\alpha = \prod_{n \rightarrow \alpha} G_n}, defined as the ultraproduct of finite {K}-approximate groups {A_n \subset G_n} for some standard {K}. (A {K}-approximate group {A_n} is a symmetric set containing the origin such that {A_n+A_n} can be covered by {K} or fewer translates of {A_n}.) We then let {O(A_\alpha)} be the external subgroup of {G_\alpha} generated by {A_\alpha}; equivalently, {A_\alpha} is the union of {A_\alpha^m} over all standard {m}. This space has a Loeb measure {\mu_{O(A_\alpha)}}, defined by setting

\displaystyle \mu_{O(A_\alpha)}(E_\alpha) := \hbox{st} \frac{|E_\alpha|}{|A_\alpha|}

whenever {E_\alpha} is an internal subset of {A_\alpha^m} for any standard {m}, and extended to a countably additive measure; the arguments in Section 6 of this previous blog post can be easily modified to give a construction of this measure.

The Loeb measure {\mu_{O(A_\alpha)}} is a translation invariant measure on {O(A_{\alpha})}, normalised so that {A_\alpha} has Loeb measure one. As such, one should think of {O(A_\alpha)} as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that {O(A_\alpha)} is not actually a locally compact group with Haar measure, for two reasons:

  • There is not an obvious topology on {O(A_\alpha)} that makes it simultaneously locally compact, Hausdorff, and {\sigma}-compact. (One can get one or two out of three without difficulty, though.)
  • The addition operation {+\colon O(A_\alpha) \times O(A_\alpha) \rightarrow O(A_\alpha)} is not measurable from the product Loeb algebra {{\mathcal L}_{O(A_\alpha)} \times {\mathcal L}_{O(A_\alpha)}} to {{\mathcal L}_{O(\alpha)}}. Instead, it is measurable from the coarser Loeb algebra {{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}} to {{\mathcal L}_{O(\alpha)}} (compare with the analogous situation for nonstandard graphs).

Nevertheless, the analogy is a useful guide for the arguments that follow.

Let {L(O(A_\alpha))} denote the space of bounded Loeb measurable functions {f\colon O(A_\alpha) \rightarrow {\bf C}} (modulo almost everywhere equivalence) that are supported on {A_\alpha^m} for some standard {m}; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation {\star\colon L(O(A_\alpha)) \times L(O(A_\alpha)) \rightarrow L(O(A_\alpha))}, defined by setting

\displaystyle  \hbox{st} f \star \hbox{st} g(x) := \hbox{st} \frac{1}{|A_\alpha|} \sum_{y \in A_\alpha^m} f(y) g(x-y)

whenever {f\colon A_\alpha^m \rightarrow {}^* {\bf C}}, {g\colon A_\alpha^l \rightarrow {}^* {\bf C}} are bounded nonstandard functions (extended by zero to all of {O(A_\alpha)}), and then extending to arbitrary elements of {L(O(A_\alpha))} by density. Equivalently, {f \star g} is the pushforward of the {{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}}-measurable function {(x,y) \mapsto f(x) g(y)} under the map {(x,y) \mapsto x+y}.

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor) Let {A_\alpha} be an ultra approximate group. Then there exists a (standard) locally compact abelian group {G} of the form

\displaystyle  G = {\bf R}^d \times {\bf Z}^m \times T

for some standard {d,m} and some compact abelian group {T}, equipped with a Haar measure {\mu_G} and a measurable homomorphism {\pi\colon O(A_\alpha) \rightarrow G} (using the Loeb {\sigma}-algebra on {O(A_\alpha)} and the Borel {\sigma}-algebra on {G}), with the following properties:

  • (i) {\pi} is surjective, and {\mu_G} is the pushforward of Loeb measure {\mu_{O(A_\alpha)}} by {\pi}.
  • (ii) There exists sets {\{0\} \subset U_0 \subset K_0 \subset G} with {U_0} open and {K_0} compact, such that

    \displaystyle  \pi^{-1}(U_0) \subset 4A_\alpha \subset \pi^{-1}(K_0). \ \ \ \ \ (1)

  • (iii) Whenever {K \subset U \subset G} with {K} compact and {U} open, there exists a nonstandard finite set {B} such that

    \displaystyle  \pi^{-1}(K) \subset B \subset \pi^{-1}(U). \ \ \ \ \ (2)

  • (iv) If {f, g \in L}, then we have the convolution formula

    \displaystyle  f \star g = \pi^*( (\pi_* f) \star (\pi_* g) ) \ \ \ \ \ (3)

    where {\pi_* f,\pi_* g} are the pushforwards of {f,g} to {L^2(G, \mu_G)}, the convolution {\star} on the right-hand side is convolution using {\mu_G}, and {\pi^*} is the pullback map from {L^2(G,\mu_G)} to {L^2(O(A_\alpha), \mu_{O(A_\alpha)})}. In particular, if {\pi_* f = 0}, then {f*g=0} for all {g \in L}.

One can view the locally compact abelian group {G} as a “model “or “Kronecker factor” for the ultra approximate group {A_\alpha} (in close analogy with the Kronecker factor from ergodic theory). In the case that {A_\alpha} is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components {{\bf R}^d \times {\bf Z}^m} of the Kronecker group {G} are trivial, and this theorem was implicitly established by Szegedy. The compact group {T} is quite large, and in particular is likely to be inseparable; but as with the case of graphons, when one is only studying at most countably many functions {f}, one can cut down the size of this group to be separable (or equivalently, second countable or metrisable) if desired, so one often works with a “reduced Kronecker factor” which is a quotient of the full Kronecker factor {G}.

Given any sequence of uniformly bounded functions {f_n\colon A_n^m \rightarrow {\bf C}} for some fixed {m}, we can view the function {f \in L} defined by

\displaystyle  f := \pi_* \hbox{st} \lim_{n \rightarrow \alpha} f_n \ \ \ \ \ (4)

as an “additive limit” of the {f_n}, in much the same way that graphons {p\colon V_\alpha \times V_\alpha \rightarrow [0,1]} are limits of the indicator functions {1_{E_n}\colon V_n \times V_n \rightarrow \{0,1\}}. The additive limits capture some of the statistics of the {f_n}, for instance the normalised means

\displaystyle  \frac{1}{|A_n|} \sum_{x \in A_n^m} f_n(x)

converge (along the ultrafilter {\alpha}) to the mean

\displaystyle  \int_G f(x)\ d\mu_G(x),

and for three sequences {f_n,g_n,h_n\colon A_n^m \rightarrow {\bf C}} of functions, the normalised correlation

\displaystyle  \frac{1}{|A_n|^2} \sum_{x,y \in A_n^m} f_n(x) g_n(y) h_n(x+y)

converges along {\alpha} to the correlation

\displaystyle  \int_G \int_G f(x) g(y) h(x+y)\ d\mu_G(x) d\mu_G(y),

the normalised {U^2} Gowers norm

\displaystyle  ( \frac{1}{|A_n|^3} \sum_{x,y,z,w \in A_n^m: x+w=y+z} f_n(x) \overline{f_n(y)} \overline{f_n(z)} f_n(w))^{1/4}

converges along {\alpha} to the {U^2} Gowers norm

\displaystyle  ( \int_{G \times G \times G} f(x) \overline{f(y)} \overline{f(z)} f_n(x+y-z)\ d\mu_G(x) d\mu_G(y) d\mu_G(z))^{1/4}

and so forth. We caution however that some correlations that involve evaluating more than one function at the same point will not necessarily be preserved in the additive limit; for instance the normalised {\ell^2} norm

\displaystyle  (\frac{1}{|A_n|} \sum_{x \in A_n^m} |f_n(x)|^2)^{1/2}

does not necessarily converge to the {L^2} norm

\displaystyle  (\int_G |f(x)|^2\ d\mu_G(x))^{1/2},

but can converge instead to a larger quantity, due to the presence of the orthogonal projection {\pi_*} in the definition (4) of {f}.

An important special case of an additive limit occurs when the functions {f_n\colon A_n^m \rightarrow {\bf C}} involved are indicator functions {f_n = 1_{E_n}} of some subsets {E_n} of {A_n^m}. The additive limit {f \in L} does not necessarily remain an indicator function, but instead takes values in {[0,1]} (much as a graphon {p} takes values in {[0,1]} even though the original indicators {1_{E_n}} take values in {\{0,1\}}). The convolution {f \star f\colon G \rightarrow [0,1]} is then the ultralimit of the normalised convolutions {\frac{1}{|A_n|} 1_{E_n} \star 1_{E_n}}; in particular, the measure of the support of {f \star f} provides a lower bound on the limiting normalised cardinality {\frac{1}{|A_n|} |E_n + E_n|} of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset {2E_n = E_n + E_n} could contain a large number of elements which have very few ({o(|A_n|)}) representations as the sum of two elements of {E_n}, and in the limit these portions of the sumset fall outside of the support of {f \star f}. (One can think of the support of {f \star f} as describing the “essential” sumset of {2E_n = E_n + E_n}, discarding those elements that have only very few representations.) Similarly for higher convolutions of {f}. Thus one can use additive limits to partially control the growth {k E_n} of iterated sumsets of subsets {E_n} of approximate groups {A_n}, in the regime where {k} stays bounded and {n} goes to infinity.

Theorem 1 can be proven by Fourier-analytic means (combined with Freiman’s theorem from additive combinatorics), and we will do so below the fold. For now, we give some illustrative examples of additive limits.

Example 1 (Bohr sets) We take {A_n} to be the intervals {A_n := \{ x \in {\bf Z}: |x| \leq N_n \}}, where {N_n} is a sequence going to infinity; these are {2}-approximate groups for all {n}. Let {\theta} be an irrational real number, let {I} be an interval in {{\bf R}/{\bf Z}}, and for each natural number {n} let {B_n} be the Bohr set

\displaystyle  B_n := \{ x \in A^{(n)}: \theta x \hbox{ mod } 1 \in I \}.

In this case, the (reduced) Kronecker factor {G} can be taken to be the infinite cylinder {{\bf R} \times {\bf R}/{\bf Z}} with the usual Lebesgue measure {\mu_G}. The additive limits of {1_{A_n}} and {1_{B_n}} end up being {1_A} and {1_B}, where {A} is the finite cylinder

\displaystyle  A := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]\}

and {B} is the rectangle

\displaystyle  B := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]; t \in I \}.

Geometrically, one should think of {A_n} and {B_n} as being wrapped around the cylinder {{\bf R} \times {\bf R}/{\bf Z}} via the homomorphism {x \mapsto (\frac{x}{N_n}, \theta x \hbox{ mod } 1)}, and then one sees that {B_n} is converging in some normalised weak sense to {B}, and similarly for {A_n} and {A}. In particular, the additive limit predicts the growth rate of the iterated sumsets {kB_n} to be quadratic in {k} until {k|I|} becomes comparable to {1}, at which point the growth transitions to linear growth, in the regime where {k} is bounded and {n} is large.

If {\theta = \frac{p}{q}} were rational instead of irrational, then one would need to replace {{\bf R}/{\bf Z}} by the finite subgroup {\frac{1}{q}{\bf Z}/{\bf Z}} here.

Example 2 (Structured subsets of progressions) We take {A_n} be the rank two progression

\displaystyle  A_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|, |b| \leq N_n \},

where {N_n} is a sequence going to infinity; these are {4}-approximate groups for all {n}. Let {B_n} be the subset

\displaystyle  B_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|^2 + |b|^2 \leq N_n^2 \}.

Then the (reduced) Kronecker factor can be taken to be {G = {\bf R}^2} with Lebesgue measure {\mu_G}, and the additive limits of the {1_{A_n}} and {1_{B_n}} are then {1_A} and {1_B}, where {A} is the square

\displaystyle  A := \{ (a,b) \in {\bf R}^2: |a|, |b| \leq 1 \}

and {B} is the circle

\displaystyle  B := \{ (a,b) \in {\bf R}^2: a^2+b^2 \leq 1 \}.

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism {a + b N_n^2 \mapsto (\frac{a}{N_n}, \frac{b}{N_n})} for {a,b = O( N_n )} to embed the original sets {A_n, B_n} into the plane {{\bf R}^2}. In particular, one now expects the growth rate of the iterated sumsets {k A_n} and {k B_n} to be quadratic in {k}, in the regime where {k} is bounded and {n} is large.

Example 3 (Dissociated sets) Let {d} be a fixed natural number, and take

\displaystyle  A_n = \{0, v_1,\dots,v_d,-v_1,\dots,-v_d \}

where {v_1,\dots,v_d} are randomly chosen elements of a large cyclic group {{\bf Z}/p_n{\bf Z}}, where {p_n} is a sequence of primes going to infinity. These are {O(d)}-approximate groups. The (reduced) Kronecker factor {G} can (almost surely) then be taken to be {{\bf Z}^d} with counting measure, and the additive limit of {1_{A_n}} is {1_A}, where {A = \{ 0, e_1,\dots,e_d,-e_1,\dots,-e_d\}} and {e_1,\dots,e_d} is the standard basis of {{\bf Z}^d}. In particular, the growth rates of {k A_n} should grow approximately like {k^d} for {k} bounded and {n} large.

Example 4 (Random subsets of groups) Let {A_n = G_n} be a sequence of finite additive groups whose order is going to infinity. Let {B_n} be a random subset of {G_n} of some fixed density {0 \leq \lambda \leq 1}. Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group {\{0\}}, and the additive limit of the {1_{B_n}} is the constant function {\lambda}. The convolutions {\frac{1}{|G_n|} 1_{B_n} * 1_{B_n}} then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of {\lambda^2}; this reflects the fact that {(1-o(1))|G_n|} of the elements of {G_n} can be represented as the sum of two elements of {B_n} in {(\lambda^2 + o(1)) |G_n|} ways. In particular, {B_n+B_n} occupies a proportion {1-o(1)} of {G_n}.

Example 5 (Trigonometric series) Take {A_n = G_n = {\bf Z}/p_n {\bf C}} for a sequence {p_n} of primes going to infinity, and for each {n} let {\xi_{n,1},\xi_{n,2},\dots} be an infinite sequence of frequencies chosen uniformly and independently from {{\bf Z}/p_n{\bf Z}}. Let {f_n\colon {\bf Z}/p_n{\bf Z} \rightarrow {\bf C}} denote the random trigonometric series

\displaystyle  f_n(x) := \sum_{j=1}^\infty 2^{-j} e^{2\pi i \xi_{n,j} x / p_n }.

Then (almost surely) we can take the reduced Kronecker factor {G} to be the infinite torus {({\bf R}/{\bf Z})^{\bf N}} (with the Haar probability measure {\mu_G}), and the additive limit of the {f_n} then becomes the function {f\colon ({\bf R}/{\bf Z})^{\bf N} \rightarrow {\bf R}} defined by the formula

\displaystyle  f( (x_j)_{j=1}^\infty ) := \sum_{j=1}^\infty e^{2\pi i x_j}.

In fact, the pullback {\pi^* f} is the ultralimit of the {f_n}. As such, for any standard exponent {1 \leq q < \infty}, the normalised {l^q} norm

\displaystyle  (\frac{1}{p_n} \sum_{x \in {\bf Z}/p_n{\bf Z}} |f_n(x)|^q)^{1/q}

can be seen to converge to the limit

\displaystyle  (\int_{({\bf R}/{\bf Z})^{\bf N}} |f(x)|^q\ d\mu_G(x))^{1/q}.

The reader is invited to consider combinations of the above examples, e.g. random subsets of Bohr sets, to get a sense of the general case of Theorem 1.

It is likely that this theorem can be extended to the noncommutative setting, using the noncommutative Freiman theorem of Emmanuel Breuillard, Ben Green, and myself, but I have not attempted to do so here (see though this recent preprint of Anush Tserunyan for some related explorations); in a separate direction, there should be extensions that can control higher Gowers norms, in the spirit of the work of Szegedy.

Note: the arguments below will presume some familiarity with additive combinatorics and with nonstandard analysis, and will be a little sketchy in places.

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One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups).

Theorem 1 (Cayley’s theorem) Let {G} be a group of some finite order {n}. Then {G} is isomorphic to a subgroup {\tilde G} of the symmetric group {S_n} on {n} elements {\{1,\dots,n\}}. Furthermore, this subgroup is simply transitive: given two elements {x,y} of {\{1,\dots,n\}}, there is precisely one element {\sigma} of {\tilde G} such that {\sigma(x)=y}.

One can therefore think of {S_n} as a sort of “universal” group that contains (up to isomorphism) all the possible groups of order {n}.

Proof: The group {G} acts on itself by multiplication on the left, thus each element {g \in G} may be identified with a permutation {\sigma_g: G \rightarrow G} on {G} given by the map {\sigma_g(h) := gh}. This can be easily verified to identify {G} with a simply transitive permutation group on {G}. The claim then follows by arbitrarily identifying {G} with {\{1,\dots,n\}}. \Box

More explicitly, the permutation group {\tilde G} arises by arbitrarily enumerating {G} as {\{s_1,\dots,s_n\}} and then associating to each group element {g \in G} the permutation {\sigma_g: \{1,\dots,n\} \rightarrow \{1,\dots,n\}} defined by the formula

\displaystyle g s_i = s_{\sigma_g(i)}.

The simply transitive group {\tilde G} given by Cayley’s theorem is not unique, due to the arbitrary choice of identification of {G} with {\{1,\dots,n\}}, but is unique up to conjugation by an element of {S_n}. On the other hand, it is easy to see that every simply transitive subgroup of {S_n} is of order {n}, and that two such groups are isomorphic if and only if they are conjugate by an element of {S_n}. Thus Cayley’s theorem in fact identifies the moduli space of groups of order {n} (up to isomorphism) with the simply transitive subgroups of {S_n} (up to conjugacy by elements of {S_n}).

One can generalise Cayley’s theorem to groups of infinite order without much difficulty. But in this post, I would like to note an (easy) generalisation of Cayley’s theorem in a different direction, in which the group {G} is no longer assumed to be of order {n}, but rather to have an index {n} subgroup that is isomorphic to a fixed group {H}. The generalisation is:

Theorem 2 (Cayley’s theorem for {H}-sets) Let {H} be a group, and let {G} be a group that contains an index {n} subgroup isomorphic to {H}. Then {G} is isomorphic to a subgroup {\tilde G} of the semidirect product {S_n \ltimes H^n}, defined explicitly as the set of tuples {(\sigma, (h_i)_{i=1}^n)} with product

\displaystyle  (\sigma, (h_i)_{i=1}^n) (\rho, (k_i)_{i=1}^n) := (\sigma \circ \rho, (h_{\rho(i)} k_i)_{i=1}^n )

and inverse

\displaystyle  (\sigma, (h_i)_{i=1}^n)^{-1} := (\sigma^{-1}, (h_{\sigma(i)}^{-1})_{i=1}^n).

(This group is a wreath product of {H} with {S_n}, and is sometimes denoted {H \wr S_n}, or more precisely {H \wr_{\{1,\dots,n\}} S_n}.) Furthermore, {\tilde G} is simply transitive in the following sense: given any two elements {x,y} of {\{1,\dots,n\}} and {h,k \in H}, there is precisely one {(\sigma, (h_i)_{i=1}^n)} in {\tilde G} such that {\sigma(x)=y} and {k = h_x h}.

Of course, Theorem 1 is the special case of Theorem 2 when {H} is trivial. This theorem allows one to view {S_n \ltimes H^n} as a “universal” group for modeling all groups containing a copy of {H} as an index {n} subgroup, in exactly the same way that {S_n} is a universal group for modeling groups of order {n}. This observation is not at all deep, but I had not seen it before, so I thought I would record it here. (EDIT: as pointed out in comments, this is a slight variant of the universal embedding theorem of Krasner and Kaloujnine, which covers the case when {H} is normal, in which case one can embed {G} into the wreath product {H \wr G/H}, which is a subgroup of {H \wr S_n}.)

Proof: The basic idea here is to replace the category of sets in Theorem 1 by the category of {H}-sets, by which we mean sets {X} with a right-action of the group {H}. A morphism between two {H}-sets {X,Y} is a function {f: X \rightarrow Y} which respects the right action of {H}, thus {f(x)h = f(xh)} for all {x \in X} and {h \in H}.

Observe that if {G} contains a copy of {H} as a subgroup, then one can view {G} as an {H}-set, using the right-action of {H} (which we identify with the indicated subgroup of {G}). The left action of {G} on itself commutes with the right-action of {H}, and so we can represent {G} by {H}-set automorphisms on the {H}-set {G}.

As {H} has index {n} in {G}, we see that {G} is (non-canonically) isomorphic (as an {H}-set) to the {H}-set {\{1,\dots,n\} \times H} with the obvious right action of {H}: {(i,h) k := (i,hk)}. It is easy to see that the group of {H}-set automorphisms of {\{1,\dots,n\} \times H} can be identified with {S^n \ltimes H}, with the latter group acting on the former {H}-set by the rule

\displaystyle  (\sigma, (h_i)_{i=1}^n) (i,h) := (\sigma(i), h_i h)

(it is routine to verify that this is indeed an action of {S^n \ltimes H} by {H}-set automorphisms. It is then a routine matter to verify the claims (the simple transitivity of {\tilde G} follows from the simple transitivity of the action of {G} on itself). \Box

More explicitly, the group {\tilde G} arises by arbitrarily enumerating the left-cosets of {H} in {G} as {\{s_1H,\dots,s_nH\}} and then associating to each group element {g \in G} the element {(\sigma_g, (h_{g,i})_{i=1}^n )}, where the permutation {\sigma_g: \{1,\dots,n\} \rightarrow \{1,\dots,n\}} and the elements {h_{g,i} \in H} are defined by the formula

\displaystyle  g s_i = s_{\sigma_g(i)} h_{g,i}.

By noting that {H^n} is an index {n!} normal subgroup of {S_n \ltimes H^n}, we recover the classical result of Poincaré that any group {G} that contains {H} as an index {n} subgroup, contains a normal subgroup {N} of index dividing {n!} that is contained in {H}. (Quotienting out the {H} right-action, we recover also the classical proof of this result, as the action of {G} on itself then collapses to the action of {G} on the quotient space {G/H}, the stabiliser of which is {N}.)

Exercise 1 Show that a simply transitive subgroup {\tilde G} of {S_n \ltimes H^n} contains a copy of {H} as an index {n} subgroup; in particular, there is a canonical embedding of {H} into {\tilde G}, and {\tilde G} can be viewed as an {H}-set.

Exercise 2 Show that any two simply transitive subgroups {\tilde G_1, \tilde G_2} of {S_n \ltimes H^n} are isomorphic simultaneously as groups and as {H}-sets (that is, there is a bijection {\phi: \tilde G_1 \rightarrow \tilde G_2} that is simultaneously a group isomorphism and an {H}-set isomorphism) if and only if they are conjugate by an element of {S_n \times H_n}.

[UPDATE: Exercises corrected; thanks to Keith Conrad for some additional corrections and comments.]

Due to some requests, I’m uploading to my blog the slides for my recent talk in Segovia (for the birthday conference of Michael Cowling) on “Hilbert’s fifth problem and approximate groups“.  The slides cover essentially the same range of topics in this series of lecture notes, or in this text of mine, though of course in considerably less detail, given that the slides are meant to be presented in an hour.

The classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space {(\Omega, {\cal E}, {\bf P})} – a space {\Omega} (the sample space) equipped with a {\sigma}-algebra {{\cal E}} (the event space), together with a countably additive probability measure {{\bf P}: {\cal E} \rightarrow [0,1]} that assigns a real number in the interval {[0,1]} to each event.

One can generalise the concept of a probability space to a finitely additive probability space, in which the event space {{\cal E}} is now only a Boolean algebra rather than a {\sigma}-algebra, and the measure {\mu} is now only finitely additive instead of countably additive, thus {{\bf P}( E \vee F ) = {\bf P}(E) + {\bf P}(F)} when {E,F} are disjoint events. By giving up countable additivity, one loses a fair amount of measure and integration theory, and in particular the notion of the expectation of a random variable becomes problematic (unless the random variable takes only finitely many values). Nevertheless, one can still perform a fair amount of probability theory in this weaker setting.

In this post I would like to describe a further weakening of probability theory, which I will call qualitative probability theory, in which one does not assign a precise numerical probability value {{\bf P}(E)} to each event, but instead merely records whether this probability is zero, one, or something in between. Thus {{\bf P}} is now a function from {{\cal E}} to the set {\{0, I, 1\}}, where {I} is a new symbol that replaces all the elements of the open interval {(0,1)}. In this setting, one can no longer compute quantitative expressions, such as the mean or variance of a random variable; but one can still talk about whether an event holds almost surely, with positive probability, or with zero probability, and there are still usable notions of independence. (I will refer to classical probability theory as quantitative probability theory, to distinguish it from its qualitative counterpart.)

The main reason I want to introduce this weak notion of probability theory is that it becomes suited to talk about random variables living inside algebraic varieties, even if these varieties are defined over fields other than {{\bf R}} or {{\bf C}}. In algebraic geometry one often talks about a “generic” element of a variety {V} defined over a field {k}, which does not lie in any specified variety of lower dimension defined over {k}. Once {V} has positive dimension, such generic elements do not exist as classical, deterministic {k}-points {x} in {V}, since of course any such point lies in the {0}-dimensional subvariety {\{x\}} of {V}. There are of course several established ways to deal with this problem. One way (which one might call the “Weil” approach to generic points) is to extend the field {k} to a sufficiently transcendental extension {\tilde k}, in order to locate a sufficient number of generic points in {V(\tilde k)}. Another approach (which one might dub the “Zariski” approach to generic points) is to work scheme-theoretically, and interpret a generic point in {V} as being associated to the zero ideal in the function ring of {V}. However I want to discuss a third perspective, in which one interprets a generic point not as a deterministic object, but rather as a random variable {{\bf x}} taking values in {V}, but which lies in any given lower-dimensional subvariety of {V} with probability zero. This interpretation is intuitive, but difficult to implement in classical probability theory (except perhaps when considering varieties over {{\bf R}} or {{\bf C}}) due to the lack of a natural probability measure to place on algebraic varieties; however it works just fine in qualitative probability theory. In particular, the algebraic geometry notion of being “generically true” can now be interpreted probabilistically as an assertion that something is “almost surely true”.

It turns out that just as qualitative random variables may be used to interpret the concept of a generic point, they can also be used to interpret the concept of a type in model theory; the type of a random variable {x} is the set of all predicates {\phi(x)} that are almost surely obeyed by {x}. In contrast, model theorists often adopt a Weil-type approach to types, in which one works with deterministic representatives of a type, which often do not occur in the original structure of interest, but only in a sufficiently saturated extension of that structure (this is the analogue of working in a sufficiently transcendental extension of the base field). However, it seems that (in some cases at least) one can equivalently view types in terms of (qualitative) random variables on the original structure, avoiding the need to extend that structure. (Instead, one reserves the right to extend the sample space of one’s probability theory whenever necessary, as part of the “probabilistic way of thinking” discussed in this previous blog post.) We illustrate this below the fold with two related theorems that I will interpret through the probabilistic lens: the “group chunk theorem” of Weil (and later developed by Hrushovski), and the “group configuration theorem” of Zilber (and again later developed by Hrushovski). For sake of concreteness we will only consider these theorems in the theory of algebraically closed fields, although the results are quite general and can be applied to many other theories studied in model theory.

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Emmanuel Breuillard, Ben Green, Bob Guralnick, and I have just uploaded to the arXiv our joint paper “Expansion in finite simple groups of Lie type“. This long-delayed paper (announced way back in 2010!) is a followup to our previous paper in which we showed that, with one possible exception, generic pairs of elements of a simple algebraic group (over an uncountable field) generated a free group which was strongly dense in the sense that any nonabelian subgroup of this group was Zariski dense. The main result of this paper is to establish the analogous result for finite simple groups of Lie type (as defined in the previous blog post) and bounded rank, namely that almost all pairs {a,b} of elements of such a group generate a Cayley graph which is a (two-sided) expander, with expansion constant bounded below by a quantity depending on the rank of the group. (Informally, this means that the random walk generated by {a,b} spreads out in logarithmic time to be essentially uniformly distributed across the group, as opposed for instance to being largely trapped in an algebraic subgroup. Thus if generic elements did not generate a strongly dense group, one would probably expect expansion to fail.)

There are also some related results established in the paper. Firstly, as we discovered after writing our first paper, there was one class of algebraic groups for which our demonstration of strongly dense subgroups broke down, namely the {Sp_4} groups in characteristic three. In the current paper we provide in a pair of appendices a new argument that covers this case (or more generally, {Sp_4} in odd characteristic), by first reducing to the case of affine groups {k^2 \rtimes SL_2(k)} (which can be found inside {Sp_4} as a subgroup) and then using a ping-pong argument (in a p-adic metric) in the latter context.

Secondly, we show that the distinction between one-sided expansion and two-sided expansion (see this set of lecture notes of mine for definitions) is erased in the context of Cayley graphs of bounded degree, in the sense that such graphs are one-sided expanders if and only if they are two-sided expanders (perhaps with slightly different expansion constants). The argument turns out to be an elementary combinatorial one, based on the “pivot” argument discussed in these lecture notes of mine.

Now to the main result of the paper, namely the expansion of random Cayley graphs. This result had previously been established for {SL_2} by Bourgain and Gamburd, and Ben, Emmanuel and I had used the Bourgain-Gamburd method to achieve the same result for Suzuki groups. For the other finite simple groups of Lie type, expander graphs had been constructed by Kassabov, Lubotzky, and Nikolov, but they required more than two generators, which were placed deterministically rather than randomly. (Here, I am skipping over a large number of other results on expanding Cayley graphs; see this survey of Lubotzsky for a fairly recent summary of developments.) The current paper also uses the “Bourgain-Gamburd machine”, as discussed in these lecture notes of mine, to demonstrate expansion. This machine shows how expansion of a Cayley graph follows from three basic ingredients, which we state informally as follows:

  • Non-concentration (A random walk in this graph does not concentrate in a proper subgroup);
  • Product theorem (A medium-sized subset of this group which is not trapped in a proper subgroup will expand under multiplication); and
  • Quasirandomness (The group has no small non-trivial linear representations).

Quasirandomness of arbitrary finite simple groups of Lie type was established many years ago (predating, in fact, the introduction of the term “quasirandomness” by Gowers for this property) by Landazuri-Seitz and Seitz-Zalesskii, and the product theorem was already established by Pyber-Szabo and independently by Breuillard, Green, and myself. So the main problem is to establish non-concentration: that for a random Cayley graph on a finite simple group {G} of Lie type, random walks did not concentrate in proper subgroups.

The first step was to classify the proper subgroups of {G}. Fortunately, these are all known; in particular, such groups are either contained in proper algebraic subgroups of the algebraic group containing {G} (or a bounded cover thereof) with bounded complexity, or are else arising (up to conjugacy) from a version {G(F')} of the same group {G =G(F)} associated to a proper subfield {F'} of the field {F} respectively; this follows for instance from the work of Larsen and Pink, but also can be deduced using the classification of finite simple groups, together with some work of Aschbacher, Liebeck-Seitz, and Nori. We refer to the two types of subgroups here as “structural subgroups” and “subfield subgroups”.

To preclude concentration in a structural subgroup, we use our previous result that generic elements of an algebraic group generate a strongly dense subgroup, and so do not concentrate in any algebraic subgroup. To translate this result from the algebraic group setting to the finite group setting, we need a Schwarz-Zippel lemma for finite simple groups of Lie type. This is straightforward for Chevalley groups, but turns out to be a bit trickier for the Steinberg and Suzuki-Ree groups, and we have to go back to the Chevalley-type parameterisation of such groups in terms of (twisted) one-parameter subgroups, that can be found for instance in the text of Carter; this “twisted Schwartz-Zippel lemma” may possibly have further application to analysis on twisted simple groups of Lie type. Unfortunately, the Schwartz-Zippel estimate becomes weaker in twisted settings, and particularly in the case of triality groups {{}^3 D_4(q)}, which require a somewhat ad hoc additional treatment that relies on passing to a simpler subgroup present in a triality group, namely a central product of two different {SL_2}‘s.

To rule out concentration in a conjugate of a subfield group, we repeat an argument we introduced in our Suzuki paper and pass to a matrix model and analyse the coefficients of the characteristic polynomial of words in this Cayley graph, to prevent them from concentrating in a subfield. (Note that these coefficients are conjugation-invariant.)

In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form {A_n}, {B_n}, {C_n}, {D_n}, {E_6}, {E_7}, {E_8}, {F_4}, or {G_2}).

Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field {{\bf C}} (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).

In the case of complex Lie groups, it turns out that every simple Lie algebra {\mathfrak{g}} is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group {G_{ad}} (the adjoint form of the Lie group) to a “maximal” Lie group {\tilde G} (the simply connected form of the Lie group) that finitely covers {G_{ad}}, and occasionally also a number of intermediate forms which finitely cover {G_{ad}}, but are in turn finitely covered by {\tilde G}. For instance, {\mathfrak{sl}_n({\bf C})} is associated with the projective special linear group {\hbox{PSL}_n({\bf C}) = \hbox{PGL}_n({\bf C})} as its adjoint form and the special linear group {\hbox{SL}_n({\bf C})} as its simply connected form, and intermediate groups can be created by quotienting out {\hbox{SL}_n({\bf C})} by some subgroup of its centre (which is isomorphic to the {n^{th}} roots of unity). The minimal form {G_{ad}} is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as simple Lie groups.

Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields {k}; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over {k} (known as Chevalley groups) with that Lie algebra, ranging from an adjoint form {G_{ad}} to a universal form {G_u}, with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form {E_7(q)_u} of the {E_7} algebraic group over a finite field {{\bf F}_q} of finite order.

When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. {\hbox{PSL}_n({\bf F}_q)}), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups {\hbox{PSU}_n({\bf F}_{q^2})}, as well as some more exotic examples such as the Suzuki groups or the Ree groups.

While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.

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A finite group {G=(G,\cdot)} is said to be a Frobenius group if there is a non-trivial subgroup {H} of {G} (known as the Frobenius complement of {G}) such that the conjugates {gHg^{-1}} of {H} are “disjoint as possible” in the sense that {H \cap gHg^{-1} = \{1\}} whenever {g \not \in H}. This gives a decomposition

\displaystyle  G = \bigcup_{gH \in G/H} (gHg^{-1} \backslash \{1\}) \cup K \ \ \ \ \ (1)

where the Frobenius kernel {K} of {G} is defined as the identity element {1} together with all the non-identity elements that are not conjugate to any element of {H}. Taking cardinalities, we conclude that

\displaystyle  |G| = \frac{|G|}{|H|} (|H| - 1) + |K|

and hence

\displaystyle  |H| |K| = |G|. \ \ \ \ \ (2)

A remarkable theorem of Frobenius gives an unexpected amount of structure on {K} and hence on {G}:

Theorem 1 (Frobenius’ theorem) Let {G} be a Frobenius group with Frobenius complement {H} and Frobenius kernel {K}. Then {K} is a normal subgroup of {G}, and hence (by (2) and the disjointness of {H} and {K} outside the identity) {G} is the semidirect product {K \rtimes H} of {H} and {K}.

I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group {G}, in particular relying on the fact that a character on a subgroup {H} can induce a character on {G}, which can then be decomposed into irreducible characters with natural number coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement {H} has even order or when {H} is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)

I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra {{\bf C} G \equiv L^2(G)}) with the Fourier analysis of class functions (i.e. the representation theory of the centre {Z({\bf C} G) \equiv L^2(G)^G} of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.

The main ideas are as follows. The space {L^2(G)^G} of class functions can be viewed as a commutative algebra with respect to the convolution operation {*}; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions {\phi} such that {\phi*\phi = \phi}. In terms of characters, idempotents are nothing more than sums of the form {\sum_{\chi \in \Sigma} \chi(1) \chi} for various collections {\Sigma} of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that {\phi(1)} is a natural number, and more generally we will show that {{\bf E}_{(a,b) \in S} {\bf E}_{x \in G} \phi( a x b^{-1} x^{-1} )} is a natural number whenever {S} is a subgroup of {G \times G} (see Corollary 4 below). For instance, the quantity

\displaystyle  \hbox{rank}(\phi) := {\bf E}_{a \in G} {\bf E}_{x \in G} \phi(a xa^{-1} x^{-1})

is a natural number which we will call the rank of {\phi} (as it is also the linear rank of the transformation {f \mapsto f*\phi} on {L^2(G)}).

In the case that {G} is a Frobenius group with kernel {K}, the above integrality properties can be used after some elementary manipulations to establish that for any idempotent {\phi}, the quantity

\displaystyle  \frac{1}{|G|} \sum_{a \in K} {\bf E}_{x \in G} \phi( axa^{-1}x^{-1} ) - \frac{1}{|G| |K|} \sum_{a,b \in K} \phi(ab^{-1}) \ \ \ \ \ (3)

is an integer. On the other hand, one can also show by elementary means that this quantity lies between {0} and {\hbox{rank}(\phi)}. These two facts are not strong enough on their own to impose much further structure on {\phi}, unless one restricts attention to minimal idempotents {\phi}. In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that {\phi} has rank one, and then the integrality gap comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent {\phi} either preserves {\frac{|G|}{|K|} 1_K} or annihilates it, which makes {\frac{|G|}{|K|} 1_K} itself an idempotent, which makes {K} normal.

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Suppose that {G = (G,\cdot)} is a finite group of even order, thus {|G|} is a multiple of two. By Cauchy’s theorem, this implies that {G} contains an involution: an element {g} in {G} of order two. (Indeed, if no such involution existed, then {G} would be partitioned into doubletons {\{g,g^{-1}\}} together with the identity, so that {|G|} would be odd, a contradiction.) Of course, groups of odd order have no involutions {g}, thanks to Lagrange’s theorem (since {G} cannot split into doubletons {\{ h, hg \}}).

The classical Brauer-Fowler theorem asserts that if a group {G} has many involutions, then it must have a large non-trivial subgroup:

Theorem 1 (Brauer-Fowler theorem) Let {G} be a finite group with at least {|G|/n} involutions for some {n > 1}. Then {G} contains a proper subgroup {H} of index at most {n^2}.

This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen {n^2} slightly to {n(n+2)/2}) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if {g} is a an involution of a group {G}, and the centraliser {C_G(g) := \{ h \in G: gh = hg\}} has order {n}, then clearly {n \geq 2} (as {C_G(g)} contains {1} and {g}) and the conjugacy class {\{ aga^{-1}: a \in G \}} has order {|G|/n} (since the map {a \mapsto aga^{-1}} has preimages that are cosets of {C_G(g)}). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem {G} contains a subgroup of order at least {\max( n, |G|/n^2)}. In particular, we can conclude that every group {G} of even order contains a proper subgroup of order at least {|G|^{1/3}}.

Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions:

Corollary 2 (Brauer-Fowler theorem) Let {G} be a finite simple group with an involution {g}, and suppose that {C_G(g)} has order {n}. Then {G} has order at most {(n^2)!}.

Indeed, by the previous discussion {G} has a proper subgroup {H} of index less than {n^2}, which then gives a non-trivial permutation action of {G} on the coset space {G/H}. The kernel of this action is a proper normal subgroup of {G} and is thus trivial, so the action is faithful, and the claim follows.

If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample {G} of minimal order {|G|} to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution {g}. Take such an involution and consider its centraliser {C_G(g)}; this is a proper subgroup of {G} of some order {n < |G|}. As {G} is a minimal counterexample to the classification, one can in principle describe {C_G(g)} in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of {C_G(g)}, that all the possible simple groups {G} that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for {C_G(g)}, there are only finitely many {G} that could generate this class for one of its centralisers, so this task should be doable in principle for any given isomorphism class for {C_G(g)}. That’s all one needs to do to prove the classification of finite simple groups!

Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a {p}-group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of {p}-groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these {p}-groups in the key case when {G} is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic {p}. See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy.

The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of quasirandomness for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2:

Corollary 3 (Weak Brauer-Fowler theorem) Let {G} be a finite simple group with an involution {g}, and suppose that {C_G(g)} has order {n}. Then {G} can be identified with a subgroup of the unitary group {U_{4n^3}({\bf C})}.

One can get an upper bound on {|G|} from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!).

Proof: Let {A} be the set of all involutions in {G}, then as discussed above {|A| \geq |G|/n}. We may assume that {G} has no non-trivial unitary representation of dimension less than {4n^3} (since such representations are automatically faithful by the simplicity of {G}); thus, in the language of quasirandomness, {G} is {4n^3}-quasirandom, and is also non-abelian. We have the basic convolution estimate

\displaystyle  \|1_A * 1_A * 1_A - \frac{|A|^3}{|G|} \|_{\ell^\infty(G)} \leq (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2}

(see Exercise 10 from this previous blog post). In particular,

\displaystyle  1_A * 1_A * 1_A(0) \geq \frac{|A|^3}{|G|} - (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2} \geq \frac{1}{2n^3} |G|^2

and so there are at least {|G|^2/2n^3} pairs {(g,h) \in A \times A} such that {gh \in A^{-1} = A}, i.e. involutions {g,h} whose product is also an involution. But any such involutions necessarily commute, since

\displaystyle  g (gh) h = g^2 h^2 = 1 = (gh)^2 = g (hg) h.

Thus there are at least {|G|^2/2n^3} pairs {(g,h) \in G \times G} of non-identity elements that commute, so by the pigeonhole principle there is a non-identity {g \in G} whose centraliser {C_G(g)} has order at least {|G|/2n^3}. This centraliser cannot be all of {G} since this would make {g} central which contradicts the non-abelian simple nature of {G}. But then the quasiregular representation of {G} on {G/C_G(g)} has dimension at most {2n^3}, contradicting the quasirandomness. \Box

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The classification of finite simple groups (CFSG), first announced in 1983 but only fully completed in 2004, is one of the monumental achievements of twentieth century mathematics. Spanning hundreds of papers and tens of thousands of pages, it has been called the “enormous theorem”. A “second generation” proof of the theorem is nearly completed which is a little shorter (estimated at about five thousand pages in length), but currently there is no reasonably sized proof of the classification.

An important precursor of the CFSG is the Feit-Thompson theorem from 1962-1963, which asserts that every finite group of odd order is solvable, or equivalently that every non-abelian finite simple group has even order. This is an immediate consequence of CFSG, and conversely the Feit-Thompson theorem is an essential starting point in the proof of the classification, since it allows one to reduce matters to groups of even order for which key additional tools (such as the Brauer-Fowler theorem) become available. The original proof of the Feit-Thompson theorem is 255 pages long, which is significantly shorter than the proof of the CFSG, but still far from short. While parts of the proof of the Feit-Thompson theorem have been simplified (and it has recently been converted, after six years of effort, into an argument that has been verified by the proof assistant Coq), the available proofs of this theorem are still extremely lengthy by any reasonable standard.

However, there is a significantly simpler special case of the Feit-Thompson theorem that was established previously by Suzuki in 1957, which was influential in the proof of the more general Feit-Thompson theorem (and thus indirectly to the proof of CFSG). Define a CA-group to be a group {G} with the property that the centraliser {C_G(x) := \{ g \in G: gx=xg \}} of any non-identity element {x \in G} is abelian; equivalently, the commuting relation {x \sim y} (defined as the relation that holds when {x} commutes with {y}, thus {xy=yx}) is an equivalence relation on the non-identity elements {G \backslash \{1\}} of {G}. Trivially, every abelian group is CA. A non-abelian example of a CA-group is the {ax+b} group of invertible affine transformations {x \mapsto ax+b} on a field {F}. A little less obviously, the special linear group {SL_2(F_q)} over a finite field {F_q} is a CA-group when {q} is a power of two. The finite simple groups of Lie type are not, in general, CA-groups, but when the rank is bounded they tend to behave as if they were “almost CA”; the centraliser of a generic element in {SL_d(F_q)}, for instance, when {d} is bounded and {q} is large), is typically a maximal torus (because most elements in {SL_d(F_q)} are regular semisimple) which is certainly abelian. In view of the CFSG, we thus see that CA or nearly CA groups form an important subclass of the simple groups, and it is thus of interest to study them separately. To this end, we have

Theorem 1 (Suzuki’s theorem on CA-groups) Every finite CA-group of odd order is solvable.

Of course, this theorem is superceded by the more general Feit-Thompson theorem, but Suzuki’s proof is substantially shorter (the original proof is nine pages) and will be given in this post. (See this survey of Solomon for some discussion of the link between Suzuki’s argument and the Feit-Thompson argument.) Suzuki’s analysis can be pushed further to give an essentially complete classification of all the finite CA-groups (of either odd or even order), but we will not pursue these matters here.

Moving even further down the ladder of simple precursors of CSFG is the following theorem of Frobenius from 1901. Define a Frobenius group to be a finite group {G} which has a subgroup {H} (called the Frobenius complement) with the property that all the non-trivial conjugates {gHg^{-1}} of {H} for {g \in G \backslash H}, intersect {H} only at the origin. For instance the {ax+b} group is also a Frobenius group (take {H} to be the affine transformations that fix a specified point {x_0 \in F}, e.g. the origin). This example suggests that there is some overlap between the notions of a Frobenius group and a CA group. Indeed, note that if {G} is a CA-group and {H} is a maximal abelian subgroup of {G}, then any conjugate {gHg^{-1}} of {H} that is not identical to {H} will intersect {H} only at the origin (because {H} and each of its conjugates consist of equivalence classes under the commuting relation {\sim}, together with the identity). So if a maximal abelian subgroup {H} of a CA-group is its own normaliser (thus {N(H) := \{ g \in G: gH=Hg\}} is equal to {H}), then the group is a Frobenius group.

Frobenius’ theorem places an unexpectedly strong amount of structure on a Frobenius group:

Theorem 2 (Frobenius’ theorem) Let {G} be a Frobenius group with Frobenius complement {H}. Then there exists a normal subgroup {K} of {G} (called the Frobenius kernel of {G}) such that {G} is the semi-direct product {H \ltimes K} of {H} and {K}.

Roughly speaking, this theorem indicates that all Frobenius groups “behave” like the {ax+b} example (which is a quintessential example of a semi-direct product).

Note that if every CA-group of odd order was either Frobenius or abelian, then Theorem 2 would imply Theorem 1 by an induction on the order of {G}, since any subgroup of a CA-group is clearly again a CA-group. Indeed, the proof of Suzuki’s theorem does basically proceed by this route (Suzuki’s arguments do indeed imply that CA-groups of odd order are Frobenius or abelian, although we will not quite establish that fact here).

Frobenius’ theorem can be reformulated in the following concrete combinatorial form:

Theorem 3 (Frobenius’ theorem, equivalent version) Let {G} be a group of permutations acting transitively on a finite set {X}, with the property that any non-identity permutation in {G} fixes at most one point in {X}. Then the set of permutations in {G} that fix no points in {X}, together with the identity, is closed under composition.

Again, a good example to keep in mind for this theorem is when {G} is the group of affine permutations on a field {F} (i.e. the {ax+b} group for that field), and {X} is the set of points on that field. In that case, the set of permutations in {G} that do not fix any points are the non-trivial translations.

To deduce Theorem 3 from Theorem 2, one applies Theorem 2 to the stabiliser of a single point in {X}. Conversely, to deduce Theorem 2 from Theorem 3, set {X := G/H = \{ gH: g \in G \}} to be the space of left-cosets of {H}, with the obvious left {G}-action; one easily verifies that this action is faithful, transitive, and each non-identity element {g} of {G} fixes at most one left-coset of {H} (basically because it lies in at most one conjugate of {H}). If we let {K} be the elements of {G} that do not fix any point in {X}, plus the identity, then by Theorem 3 {K} is closed under composition; it is also clearly closed under inverse and conjugation, and is hence a normal subgroup of {G}. From construction {K} is the identity plus the complement of all the {|G|/|H|} conjugates of {H}, which are all disjoint except at the identity, so by counting elements we see that

\displaystyle |K| = |G| - \frac{|G|}{|H|}(|H|-1) = |G|/|H|.

As {H} normalises {K} and is disjoint from {K}, we thus see that {KH = H \ltimes K} is all of {G}, giving Theorem 2.

Despite the appealingly concrete and elementary form of Theorem 3, the only known proofs of that theorem (or equivalently, Theorem 2) in its full generality proceed via the machinery of group characters (which one can think of as a version of Fourier analysis for nonabelian groups). On the other hand, once one establishes the basic theory of these characters (reviewed below the fold), the proof of Frobenius’ theorem is very short, which gives quite a striking example of the power of character theory. The proof of Suzuki’s theorem also proceeds via character theory, and is basically a more involved version of the Frobenius argument; again, no character-free proof of Suzuki’s theorem is currently known. (The proofs of Feit-Thompson and CFSG also involve characters, but those proofs also contain many other arguments of much greater complexity than the character-based portions of the proof.)

It seems to me that the above four theorems (Frobenius, Suzuki, Feit-Thompson, and CFSG) provide a ladder of sorts (with exponentially increasing complexity at each step) to the full classification, and that any new approach to the classification might first begin by revisiting the earlier theorems on this ladder and finding new proofs of these results first (in particular, if one had a “robust” proof of Suzuki’s theorem that also gave non-trivial control on “almost CA-groups” – whatever that means – then this might lead to a new route to classifying the finite simple groups of Lie type and bounded rank). But even for the simplest two results on this ladder – Frobenius and Suzuki – it seems remarkably difficult to find any proof that is not essentially the character-based proof. (Even trying to replace character theory by its close cousin, representation theory, doesn’t seem to work unless one gives in to the temptation to take traces everywhere and put the characters back in; it seems that rather than abandon characters altogether, one needs to find some sort of “robust” generalisation of existing character-based methods.) In any case, I am recording here the standard character-based proofs of the theorems of Frobenius and Suzuki below the fold. There is nothing particularly novel here, but I wanted to collect all the relevant material in one place, largely for my own benefit.

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I’ve just uploaded to the arXiv my paper “Mixing for progressions in non-abelian groups“, submitted to Forum of Mathematics, Sigma (which, along with sister publication Forum of Mathematics, Pi, has just opened up its online submission system). This paper is loosely related in subject topic to my two previous papers on polynomial expansion and on recurrence in quasirandom groups (with Vitaly Bergelson), although the methods here are rather different from those in those two papers. The starting motivation for this paper was a question posed in this foundational paper of Tim Gowers on quasirandom groups. In that paper, Gowers showed (among other things) that if {G} was a quasirandom group, patterns such as {(x,xg,xh, xgh)} were mixing in the sense that, for any four sets {A,B,C,D \subset G}, the number of such quadruples {(x,xg,xh, xgh)} in {A \times B \times C \times D} was equal to {(\mu(A) \mu(B) \mu(C) \mu(D) + o(1)) |G|^3}, where {\mu(A) := |A| / |G|}, and {o(1)} denotes a quantity that goes to zero as the quasirandomness of the group goes to infinity. In my recent paper with Vitaly, we also considered mixing properties of some other patterns, namely {(x,xg,gx)} and {(g,x,xg,gx)}. This paper is concerned instead with the pattern {(x,xg,xg^2)}, that is to say a geometric progression of length three. As observed by Gowers, by applying (a suitably quantitative version of) Roth’s theorem in (cosets of) a cyclic group, one can obtain a recurrence theorem for this pattern without much effort: if {G} is an arbitrary finite group, and {A} is a subset of {G} with {\mu(A) \geq \delta}, then there are at least {c(\delta) |G|^2} pairs {(x,g) \in G} such that {x, xg, xg^2 \in A}, where {c(\delta)>0} is a quantity depending only on {\delta}. However, this argument does not settle the question of whether there is a stronger mixing property, in that the number of pairs {(x,g) \in G^2} such that {(x,xg,xg^2) \in A \times B \times C} should be {(\mu(A)\mu(B)\mu(C)+o(1)) |G|^2} for any {A,B,C \subset G}. Informally, this would assert that for {x, g} chosen uniformly at random from {G}, the triplet {(x, xg, xg^2)} should resemble a uniformly selected element of {G^3} in some weak sense.

For non-quasirandom groups, such mixing properties can certainly fail. For instance, if {G} is the cyclic group {G = ({\bf Z}/N{\bf Z},+)} (which is abelian and thus highly non-quasirandom) with the additive group operation, and {A = \{1,\ldots,\lfloor \delta N\rfloor\}} for some small but fixed {\delta > 0}, then {\mu(A) = \delta + o(1)} in the limit {N \rightarrow \infty}, but the number of pairs {(x,g) \in G^2} with {x, x+g, x+2g \in A} is {(\delta^2/2 + o(1)) |G|^2} rather than {(\delta^3+o(1)) |G|^2}. The problem here is that the identity {(x+2g) = 2(x+g) - x} ensures that if {x} and {x+g} both lie in {A}, then {x+2g} has a highly elevated likelihood of also falling in {A}. One can view {A} as the preimage of a small ball under the one-dimensional representation {\rho: G \rightarrow U(1)} defined by {\rho(n) := e^{2\pi i n/N}}; similar obstructions to mixing can also be constructed from other low-dimensional representations.

However, by definition, quasirandom groups do not have low-dimensional representations, and Gowers asked whether mixing for {(x,xg,xg^2)} could hold for quasirandom groups. I do not know if this is the case for arbitrary quasirandom groups, but I was able to settle the question for a specific class of quasirandom groups, namely the special linear groups {G := SL_d(F)} over a finite field {F} in the regime where the dimension {d} is bounded (but is at least two) and {F} is large. Indeed, for such groups I can obtain a count of {(\mu(A) \mu(B) \mu(C) + O( |F|^{-\min(d-1,2)/8} )) |G|^2} for the number of pairs {(x,g) \in G^2} with {(x, xg, xg^2) \in A \times B \times C}. In fact, I have the somewhat stronger statement that there are {(\mu(A) \mu(B) \mu(C) \mu(D) + O( |F|^{-\min(d-1,2)/8} )) |G|^2} pairs {(x,g) \in G^2} with {(x,xg,xg^2,g) \in A \times B \times C \times D} for any {A,B,C,D \subset G}.

I was also able to obtain a partial result for the length four progression {(x,xg,xg^2, xg^3)} in the simpler two-dimensional case {G = SL_2(F)}, but I had to make the unusual restriction that the group element {g \in G} was hyperbolic in the sense that it was diagonalisable over the finite field {F} (as opposed to diagonalisable over the algebraic closure {\overline{F}} of that field); this amounts to the discriminant of the matrix being a quadratic residue, and this holds for approximately half of the elements of {G}. The result is then that for any {A,B,C,D \subset G}, one has {(\frac{1}{2} \mu(A) \mu(B) \mu(C) \mu(D) + o(1)) |G|^2} pairs {(x,g) \in G} with {g} hyperbolic and {(x,xg,xg^2,xg^3) \subset A \times B \times C \times D}. (Again, I actually show a slightly stronger statement in which {g} is restricted to an arbitrary subset {E} of hyperbolic elements.)

For the length three argument, the main tools used are the Cauchy-Schwarz inequality, the quasirandomness of {G}, and some algebraic geometry to ensure that a certain family of probability measures on {G} that are defined algebraically are approximately uniformly distributed. The length four argument is significantly more difficult and relies on a rather ad hoc argument involving, among other things, expander properties related to the work of Bourgain and Gamburd, and also a “twisted” version of an argument of Gowers that is used (among other things) to establish an inverse theorem for the {U^3} norm.

I give some details of these arguments below the fold.

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