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Suppose that are two subgroups of some ambient group , with the index of in being finite. Then is the union of left cosets of , thus for some set of cardinality . The elements of are not entirely arbitrary with regards to . For instance, if is a *normal* subgroup of , then for each , the conjugation map preserves . In particular, if we write for the conjugate of by , then

Even if is not normal in , it turns out that the conjugation map *approximately* preserves , if is bounded. To quantify this, let us call two subgroups *-commensurate* for some if one has

Proposition 1Let be groups, with finite index . Then for every , the groups and are -commensurate, in fact

*Proof:* One can partition into left translates of , as well as left translates of . Combining the partitions, we see that can be partitioned into at most non-empty sets of the form . Each of these sets is easily seen to be a left translate of the subgroup , thus . Since

and , we obtain the claim.

One can replace the inclusion by commensurability, at the cost of some worsening of the constants:

Corollary 2Let be -commensurate subgroups of . Then for every , the groups and are -commensurate.

*Proof:* Applying the previous proposition with replaced by , we see that for every , and are -commensurate. Since and have index at most in and respectively, the claim follows.

It turns out that a similar phenomenon holds for the more general concept of an *approximate group*, and gives a “classification” of all the approximate groups containing a given approximate group as a “bounded index approximate subgroup”. Recall that a -approximate group in a group for some is a symmetric subset of containing the identity, such that the product set can be covered by at most left translates of (and thus also right translates, by the symmetry of ). For simplicity we will restrict attention to finite approximate groups so that we can use their cardinality as a measure of size. We call finite two approximate groups *-commensurate* if one has

note that this is consistent with the previous notion of commensurability for genuine groups.

Theorem 3Let be a group, and let be real numbers. Let be a finite -approximate group, and let be a symmetric subset of that contains .

- (i) If is a -approximate group with , then one has for some set of cardinality at most . Furthermore, for each , the approximate groups and are -commensurate.
- (ii) Conversely, if for some set of cardinality at most , and and are -commensurate for all , then , and is a -approximate group.

Informally, the assertion that is an approximate group containing as a “bounded index approximate subgroup” is equivalent to being covered by a bounded number of shifts of , where approximately normalises in the sense that and have large intersection. Thus, to classify all such , the problem essentially reduces to that of classifying those that approximately normalise .

To prove the theorem, we recall some standard lemmas from arithmetic combinatorics, which are the foundation stones of the “Ruzsa calculus” that we will use to establish our results:

Lemma 4 (Ruzsa covering lemma)If and are finite non-empty subsets of , then one has for some set with cardinality .

*Proof:* We take to be a subset of with the property that the translates are disjoint in , and such that is maximal with respect to set inclusion. The required properties of are then easily verified.

Lemma 5 (Ruzsa triangle inequality)If are finite non-empty subsets of , then

*Proof:* If is an element of with and , then from the identity we see that can be written as the product of an element of and an element of in at least distinct ways. The claim follows.

Now we can prove (i). By the Ruzsa covering lemma, can be covered by at most

left-translates of , and hence by at most left-translates of , thus for some . Since only intersects if , we may assume that

and hence for any

By the Ruzsa covering lemma again, this implies that can be covered by at most left-translates of , and hence by at most left-translates of . By the pigeonhole principle, there thus exists a group element with

Since

and

the claim follows.

Now we prove (ii). Clearly

Now we control the size of . We have

From the Ruzsa triangle inequality and symmetry we have

and thus

By the Ruzsa covering lemma, this implies that is covered by at most left-translates of , hence by at most left-translates of . Since , the claim follows.

We now establish some auxiliary propositions about commensurability of approximate groups. The first claim is that commensurability is approximately transitive:

Proposition 6Let be a -approximate group, be a -approximate group, and be a -approximate group. If and are -commensurate, and and are -commensurate, then and are -commensurate.

*Proof:* From two applications of the Ruzsa triangle inequality we have

By the Ruzsa covering lemma, we may thus cover by at most left-translates of , and hence by left-translates of . By the pigeonhole principle, there thus exists a group element such that

and so by arguing as in the proof of part (i) of the theorem we have

and similarly

and the claim follows.

The next proposition asserts that the union and (modified) intersection of two commensurate approximate groups is again an approximate group:

Proposition 7Let be a -approximate group, be a -approximate group, and suppose that and are -commensurate. Then is a -approximate subgroup, and is a -approximate subgroup.

Using this proposition, one may obtain a variant of the previous theorem where the containment is replaced by commensurability; we leave the details to the interested reader.

*Proof:* We begin with . Clearly is symmetric and contains the identity. We have . The set is already covered by left translates of , and hence of ; similarly is covered by left translates of . As for , we observe from the Ruzsa triangle inequality that

and hence by the Ruzsa covering lemma, is covered by at most left translates of , and hence by left translates of , and hence of . Similarly is covered by at most left translates of . The claim follows.

Now we consider . Again, this is clearly symmetric and contains the identity. Repeating the previous arguments, we see that is covered by at most left-translates of , and hence there exists a group element with

Now observe that

and so by the Ruzsa covering lemma, can be covered by at most left-translates of . But this latter set is (as observed previously) contained in , and the claim follows.

Because of Euler’s identity , the complex exponential is not injective: for any complex and integer . As such, the complex logarithm is not well-defined as a single-valued function from to . However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative real axis , one has the *standard branch* of the logarithm, with defined as the unique choice of the complex logarithm of whose imaginary part has magnitude strictly less than . This particular branch has a number of useful additional properties:

- The standard branch is holomorphic on its domain .
- One has for all in the domain . In particular, if is real, then is real.
- One has for all in the domain .

One can then also use the standard branch of the logarithm to create standard branches of other multi-valued functions, for instance creating a standard branch of the square root function. We caution however that the identity can fail for the standard branch (or indeed for any branch of the logarithm).

One can extend this standard branch of the logarithm to complex matrices, or (equivalently) to linear transformations on an -dimensional complex vector space , provided that the spectrum of that matrix or transformation avoids the branch cut . Indeed, from the spectral theorem one can decompose any such as the direct sum of operators on the non-trivial generalised eigenspaces of , where ranges in the spectrum of . For each component of , we define

where is the Taylor expansion of at ; as is nilpotent, only finitely many terms in this Taylor expansion are required. The logarithm is then defined as the direct sum of the .

The matrix standard branch of the logarithm has many pleasant and easily verified properties (often inherited from their scalar counterparts), whenever has no spectrum in :

- (i) We have .
- (ii) If and have no spectrum in , then .
- (iii) If has spectrum in a closed disk in , then , where is the Taylor series of around (which is absolutely convergent in ).
- (iv) depends holomorphically on . (Easily established from (ii), (iii), after covering the spectrum of by disjoint disks; alternatively, one can use the Cauchy integral representation for a contour in the domain enclosing the spectrum of .) In particular, the standard branch of the matrix logarithm is smooth.
- (v) If is any invertible linear or antilinear map, then . In particular, the standard branch of the logarithm commutes with matrix conjugations; and if is real with respect to a complex conjugation operation on (that is to say, an antilinear involution), then is real also.
- (vi) If denotes the transpose of (with the complex dual of ), then . Similarly, if denotes the adjoint of (with the complex conjugate of , i.e. with the conjugated multiplication map ), then .
- (vii) One has .
- (viii) If denotes the spectrum of , then .

As a quick application of the standard branch of the matrix logarithm, we have

Proposition 1Let be one of the following matrix groups: , , , , , or , where is a non-degenerate real quadratic form (so is isomorphic to a (possibly indefinite) orthogonal group for some . Then any element of whose spectrum avoids is exponential, that is to say for some in the Lie algebra of .

*Proof:* We just prove this for , as the other cases are similar (or a bit simpler). If , then (viewing as a complex-linear map on , and using the complex bilinear form associated to to identify with its complex dual , then is real and . By the properties (v), (vi), (vii) of the standard branch of the matrix logarithm, we conclude that is real and , and so lies in the Lie algebra , and the claim now follows from (i).

Exercise 2Show that is not exponential in if . Thus we see that the branch cut in the above proposition is largely necessary. See this paper of Djokovic for a more complete description of the image of the exponential map in classical groups, as well as this previous blog post for some more discussion of the surjectivity (or lack thereof) of the exponential map in Lie groups.

For a slightly less quick application of the standard branch, we have the following result (recently worked out in the answers to this MathOverflow question):

Proposition 3Let be an element of the split orthogonal group which lies in the connected component of the identity. Then .

The requirement that lie in the identity component is necessary, as the counterexample for shows.

*Proof:* We think of as a (real) linear transformation on , and write for the quadratic form associated to , so that . We can split , where is the sum of all the generalised eigenspaces corresponding to eigenvalues in , and is the sum of all the remaining eigenspaces. Since and are real, are real (i.e. complex-conjugation invariant) also. For , the restriction of to then lies in , where is the restriction of to , and

The spectrum of consists of positive reals, as well as complex pairs (with equal multiplicity), so . From the preceding proposition we have for some ; this will be important later.

It remains to show that . If has spectrum at then we are done, so we may assume that has spectrum only at (being invertible, has no spectrum at ). We split , where correspond to the portions of the spectrum in , ; these are real, -invariant spaces. We observe that if are generalised eigenspaces of with , then are orthogonal with respect to the (complex-bilinear) inner product associated with ; this is easiest to see first for the actual eigenspaces (since for all ), and the extension to generalised eigenvectors then follows from a routine induction. From this we see that is orthogonal to , and and are null spaces, which by the non-degeneracy of (and hence of the restriction of to ) forces to have the same dimension as , indeed now gives an identification of with . If we let be the restrictions of to , we thus identify with , since lies in ; in particular is invertible. Thus

and so it suffices to show that .

At this point we need to use the hypothesis that lies in the identity component of . This implies (by a continuity argument) that the restriction of to any maximal-dimensional positive subspace has positive determinant (since such a restriction cannot be singular, as this would mean that positive norm vector would map to a non-positive norm vector). Now, as have equal dimension, has a balanced signature, so does also. Since , already lies in the identity component of , and so has positive determinant on any maximal-dimensional positive subspace of . We conclude that has positive determinant on any maximal-dimensional positive subspace of .

We choose a complex basis of , to identify with , which has already been identified with . (In coordinates, are now both of the form , and for .) Then becomes a maximal positive subspace of , and the restriction of to this subspace is conjugate to , so that

But since and is positive definite, so as required.

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 of finite graphs, one can extract a subsequence which converges (in a specific sense) to a continuous object known as a “graphon” – a symmetric measurable function . What “converges” means in this context is that subgraph densities converge to the associated integrals of the graphon . For instance, the edge density

converge to the integral

the triangle density

converges to the integral

the four-cycle density

converges to the integral

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 of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter ) to obtain a nonstandard graph , where is the ultraproduct of the , and similarly for the . The set can then be viewed as a symmetric subset of which is measurable with respect to the Loeb -algebra of the product (see this previous blog post for the construction of Loeb measure). A crucial point is that this -algebra is larger than the product of the Loeb -algebra of the individual vertex set . This leads to a decomposition

where the “graphon” is the orthogonal projection of onto , and the “regular error” is orthogonal to all product sets for . The graphon then captures the statistics of the nonstandard graph , in exact analogy with the more traditional graph limits: for instance, the edge density

(or equivalently, the limit of the along the ultrafilter ) is equal to the integral

where denotes Loeb measure on a nonstandard finite set ; the triangle density

(or equivalently, the limit along of the triangle densities of ) is equal to the integral

and so forth. Note that with this construction, the graphon is living on the Cartesian square of an abstract probability space , which is likely to be inseparable; but it is possible to cut down the Loeb -algebra on to minimal countable -algebra for which remains measurable (up to null sets), and then one can identify with , 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 of an abelian group , 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* in a nonstandard group , defined as the ultraproduct of finite -approximate groups for some standard . (A -approximate group is a symmetric set containing the origin such that can be covered by or fewer translates of .) We then let be the external subgroup of generated by ; equivalently, is the union of over all standard . This space has a Loeb measure , defined by setting

whenever is an internal subset of for any standard , 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 is a translation invariant measure on , normalised so that has Loeb measure one. As such, one should think of as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that is not *actually* a locally compact group with Haar measure, for two reasons:

- There is not an obvious topology on that makes it simultaneously locally compact, Hausdorff, and -compact. (One can get one or two out of three without difficulty, though.)
- The addition operation is not measurable from the product Loeb algebra to . Instead, it is measurable from the coarser Loeb algebra to (compare with the analogous situation for nonstandard graphs).

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

Let denote the space of bounded Loeb measurable functions (modulo almost everywhere equivalence) that are supported on for some standard ; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation , defined by setting

whenever , are bounded nonstandard functions (extended by zero to all of ), and then extending to arbitrary elements of by density. Equivalently, is the pushforward of the -measurable function under the map .

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor)Let be an ultra approximate group. Then there exists a (standard) locally compact abelian group of the formfor some standard and some compact abelian group , equipped with a Haar measure and a measurable homomorphism (using the Loeb -algebra on and the Baire -algebra on ), with the following properties:

- (i) has dense image, and is the pushforward of Loeb measure by .
- (ii) There exists sets with open and compact, such that
- (iii) Whenever with compact and open, there exists a nonstandard finite set such that
- (iv) If , then we have the convolution formula
where are the pushforwards of to , the convolution on the right-hand side is convolution using , and is the pullback map from to . In particular, if , then for all .

One can view the locally compact abelian group as a “model “or “Kronecker factor” for the ultra approximate group (in close analogy with the Kronecker factor from ergodic theory). In the case that is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components of the Kronecker group are trivial, and this theorem was implicitly established by Szegedy. The compact group 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 , 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 . Once one is in the separable case, the Baire sigma algebra is identical with the more familiar Borel sigma algebra.

Given any sequence of uniformly bounded functions for some fixed , we can view the function defined by

as an “additive limit” of the , in much the same way that graphons are limits of the indicator functions . The additive limits capture some of the statistics of the , for instance the normalised means

converge (along the ultrafilter ) to the mean

and for three sequences of functions, the normalised correlation

converges along to the correlation

the normalised Gowers norm

converges along to the Gowers norm

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 norm

does not necessarily converge to the norm

but can converge instead to a larger quantity, due to the presence of the orthogonal projection in the definition (4) of .

An important special case of an additive limit occurs when the functions involved are indicator functions of some subsets of . The additive limit does not necessarily remain an indicator function, but instead takes values in (much as a graphon takes values in even though the original indicators take values in ). The convolution is then the ultralimit of the normalised convolutions ; in particular, the measure of the support of provides a lower bound on the limiting normalised cardinality of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset could contain a large number of elements which have very few () representations as the sum of two elements of , and in the limit these portions of the sumset fall outside of the support of . (One can think of the support of as describing the “essential” sumset of , discarding those elements that have only very few representations.) Similarly for higher convolutions of . Thus one can use additive limits to partially control the growth of iterated sumsets of subsets of approximate groups , in the regime where stays bounded and 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 2 (Bohr sets)We take to be the intervals , where is a sequence going to infinity; these are -approximate groups for all . Let be an irrational real number, let be an interval in , and for each natural number let be the Bohr setIn this case, the (reduced) Kronecker factor can be taken to be the infinite cylinder with the usual Lebesgue measure . The additive limits of and end up being and , where is the finite cylinder

and is the rectangle

Geometrically, one should think of and as being wrapped around the cylinder via the homomorphism , and then one sees that is converging in some normalised weak sense to , and similarly for and . In particular, the additive limit predicts the growth rate of the iterated sumsets to be quadratic in until becomes comparable to , at which point the growth transitions to linear growth, in the regime where is bounded and is large.

If were rational instead of irrational, then one would need to replace by the finite subgroup here.

Example 3 (Structured subsets of progressions)We take be the rank two progressionwhere is a sequence going to infinity; these are -approximate groups for all . Let be the subset

Then the (reduced) Kronecker factor can be taken to be with Lebesgue measure , and the additive limits of the and are then and , where is the square

and is the circle

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism for to embed the original sets into the plane . In particular, one now expects the growth rate of the iterated sumsets and to be quadratic in , in the regime where is bounded and is large.

Example 4 (Dissociated sets)Let be a fixed natural number, and takewhere are randomly chosen elements of a large cyclic group , where is a sequence of primes going to infinity. These are -approximate groups. The (reduced) Kronecker factor can (almost surely) then be taken to be with counting measure, and the additive limit of is , where and is the standard basis of . In particular, the growth rates of should grow approximately like for bounded and large.

Example 5 (Random subsets of groups)Let be a sequence of finite additive groups whose order is going to infinity. Let be a random subset of of some fixed density . Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group , and the additive limit of the is the constant function . The convolutions then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of ; this reflects the fact that of the elements of can be represented as the sum of two elements of in ways. In particular, occupies a proportion of .

Example 6 (Trigonometric series)Take for a sequence of primes going to infinity, and for each let be an infinite sequence of frequencies chosen uniformly and independently from . Let denote the random trigonometric seriesThen (almost surely) we can take the reduced Kronecker factor to be the infinite torus (with the Haar probability measure ), and the additive limit of the then becomes the function defined by the formula

In fact, the pullback is the ultralimit of the . As such, for any standard exponent , the normalised norm

can be seen to converge to the limit

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.

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 be a group of some finite order . Then is isomorphic to a subgroup of the symmetric group on elements . Furthermore, this subgroup is simply transitive: given two elements of , there is precisely one element of such that .

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

*Proof:* The group acts on itself by multiplication on the left, thus each element may be identified with a permutation on given by the map . This can be easily verified to identify with a simply transitive permutation group on . The claim then follows by arbitrarily identifying with .

More explicitly, the permutation group arises by arbitrarily enumerating as and then associating to each group element the permutation defined by the formula

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

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 is no longer assumed to be of order , but rather to have an index subgroup that is isomorphic to a fixed group . The generalisation is:

Theorem 2 (Cayley’s theorem for -sets)Let be a group, and let be a group that contains an index subgroup isomorphic to . Then is isomorphic to a subgroup of the semidirect product , defined explicitly as the set of tuples with productand inverse

(This group is a wreath product of with , and is sometimes denoted , or more precisely .) Furthermore, is simply transitive in the following sense: given any two elements of and , there is precisely one in such that and .

Of course, Theorem 1 is the special case of Theorem 2 when is trivial. This theorem allows one to view as a “universal” group for modeling all groups containing a copy of as an index subgroup, in exactly the same way that is a universal group for modeling groups of order . 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 is normal, in which case one can embed into the wreath product , which is a subgroup of .)

*Proof:* The basic idea here is to replace the category of sets in Theorem 1 by the category of -sets, by which we mean sets with a right-action of the group . A morphism between two -sets is a function which respects the right action of , thus for all and .

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

As has index in , we see that is (non-canonically) isomorphic (as an -set) to the -set with the obvious right action of : . It is easy to see that the group of -set automorphisms of can be identified with , with the latter group acting on the former -set by the rule

(it is routine to verify that this is indeed an action of by -set automorphisms. It is then a routine matter to verify the claims (the simple transitivity of follows from the simple transitivity of the action of on itself).

More explicitly, the group arises by arbitrarily enumerating the left-cosets of in as and then associating to each group element the element , where the permutation and the elements are defined by the formula

By noting that is an index normal subgroup of , we recover the classical result of Poincaré that any group that contains as an index subgroup, contains a normal subgroup of index dividing that is contained in . (Quotienting out the right-action, we recover also the classical *proof* of this result, as the action of on itself then collapses to the action of on the quotient space , the stabiliser of which is .)

Exercise 1Show that a simply transitive subgroup of contains a copy of as an index subgroup; in particular, there is a canonical embedding of into , and can be viewed as an -set.

Exercise 2Show that any two simply transitive subgroups of are isomorphic simultaneously as groups and as -sets (that is, there is a bijection that is simultaneously a group isomorphism and an -set isomorphism) if and only if they are conjugate by an element of .

[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 – a space (the sample space) equipped with a -algebra (the event space), together with a countably additive probability measure that assigns a real number in the interval to each event.

One can generalise the concept of a probability space to a *finitely additive* probability space, in which the event space is now only a Boolean algebra rather than a -algebra, and the measure is now only finitely additive instead of countably additive, thus when 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 to each event, but instead merely records whether this probability is zero, one, or something in between. Thus is now a function from to the set , where is a new symbol that replaces all the elements of the open interval . 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 or . In algebraic geometry one often talks about a “generic” element of a variety defined over a field , which does not lie in any specified variety of lower dimension defined over . Once has positive dimension, such generic elements do not exist as classical, deterministic -points in , since of course any such point lies in the -dimensional subvariety of . 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 to a sufficiently transcendental extension , in order to locate a sufficient number of generic points in . Another approach (which one might dub the “Zariski” approach to generic points) is to work scheme-theoretically, and interpret a generic point in as being associated to the zero ideal in the function ring of . 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* taking values in , but which lies in any given lower-dimensional subvariety of with probability zero. This interpretation is intuitive, but difficult to implement in classical probability theory (except perhaps when considering varieties over or ) 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 is the set of all predicates that are almost surely obeyed by . 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.

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 , , , , , , , , or ).

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 (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 is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group (the *adjoint form* of the Lie group) to a “maximal” Lie group (the *simply connected form* of the Lie group) that finitely covers , and occasionally also a number of intermediate forms which finitely cover , but are in turn finitely covered by . For instance, is associated with the projective special linear group as its adjoint form and the special linear group as its simply connected form, and intermediate groups can be created by quotienting out by some subgroup of its centre (which is isomorphic to the roots of unity). The minimal form 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 ; 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 (known as *Chevalley groups*) with that Lie algebra, ranging from an adjoint form to a universal form , 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 of the algebraic group over a finite field of finite order.

When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ), 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 , 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.

A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the *Frobenius complement* of ) such that the conjugates of are “disjoint as possible” in the sense that whenever . This gives a decomposition

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

A remarkable theorem of Frobenius gives an unexpected amount of structure on and hence on :

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

I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group , in particular relying on the fact that a character on a subgroup can induce a character on , 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 has even order or when 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 ) with the Fourier analysis of class functions (i.e. the representation theory of the centre 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 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 such that . In terms of characters, idempotents are nothing more than sums of the form for various collections 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 is a natural number, and more generally we will show that is a natural number whenever is a subgroup of (see Corollary 4 below). For instance, the quantity

is a natural number which we will call the *rank* of (as it is also the linear rank of the transformation on ).

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

is an integer. On the other hand, one can also show by elementary means that this quantity lies between and . These two facts are not strong enough on their own to impose much further structure on , unless one restricts attention to *minimal* idempotents . In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that 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 either preserves or annihilates it, which makes itself an idempotent, which makes normal.

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

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

Theorem 1 (Brauer-Fowler theorem)Let be a finite group with at least involutions for some . Then contains a proper subgroup of index at most .

This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen slightly to ) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if is a an involution of a group , and the centraliser has order , then clearly (as contains and ) and the conjugacy class has order (since the map has preimages that are cosets of ). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem contains a subgroup of order at least . In particular, we can conclude that every group of even order contains a proper subgroup of order at least .

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 be a finite simple group with an involution , and suppose that has order . Then has order at most .

Indeed, by the previous discussion has a proper subgroup of index less than , which then gives a non-trivial permutation action of on the coset space . The kernel of this action is a proper normal subgroup of 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 of minimal order 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 . Take such an involution and consider its centraliser ; this is a proper subgroup of of some order . As is a minimal counterexample to the classification, one can in principle describe 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 , that all the possible simple groups 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 , there are only finitely many that could generate this class for one of its centralisers, so this task should be doable *in principle* for any given isomorphism class for . 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 -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 -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 -groups in the key case when is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic . 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 be a finite simple group with an involution , and suppose that has order . Then can be identified with a subgroup of the unitary group .

One can get an upper bound on 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 be the set of all involutions in , then as discussed above . We may assume that has no non-trivial unitary representation of dimension less than (since such representations are automatically faithful by the simplicity of ); thus, in the language of quasirandomness, is -quasirandom, and is also non-abelian. We have the basic convolution estimate

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

and so there are at least pairs such that , i.e. involutions whose product is also an involution. But any such involutions necessarily commute, since

Thus there are at least pairs of non-identity elements that commute, so by the pigeonhole principle there is a non-identity whose centraliser has order at least . This centraliser cannot be all of since this would make central which contradicts the non-abelian simple nature of . But then the quasiregular representation of on has dimension at most , contradicting the quasirandomness.

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