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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.

This is another installment of my my series of posts on Hilbert’s fifth problem. One formulation of this problem is answered by the following theorem of Gleason and Montgomery-Zippin:

Theorem 1 (Hilbert’s fifth problem) Let ${G}$ be a topological group which is locally Euclidean. Then ${G}$ is isomorphic to a Lie group.

Theorem 1 is deep and difficult result, but the discussion in the previous posts has reduced the proof of this Theorem to that of establishing two simpler results, involving the concepts of a no small subgroups (NSS) subgroup, and that of a Gleason metric. We briefly recall the relevant definitions:

Definition 2 (NSS) A topological group ${G}$ is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood ${U}$ of the identity in ${G}$ that contains no subgroups of ${G}$ other than the trivial subgroup ${\{ \hbox{id}\}}$.

Definition 3 (Gleason metric) Let ${G}$ be a topological group. A Gleason metric on ${G}$ is a left-invariant metric ${d: G \times G \rightarrow {\bf R}^+}$ which generates the topology on ${G}$ and obeys the following properties for some constant ${C>0}$, writing ${\|g\|}$ for ${d(g,\hbox{id})}$:

The remaining steps in the resolution of Hilbert’s fifth problem are then as follows:

Theorem 4 (Reduction to the NSS case) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is NSS and locally compact.

Theorem 5 (Gleason’s lemma) Let ${G}$ be a locally compact NSS group. Then ${G}$ has a Gleason metric.

The purpose of this post is to establish these two results, using arguments that are originally due to Gleason. We will split this task into several subtasks, each of which improves the structure on the group ${G}$ by some amount:

Proposition 6 (From locally compact to metrisable) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is locally compact and metrisable.

For any open neighbourhood ${U}$ of the identity in ${G}$, let ${Q(U)}$ be the union of all the subgroups of ${G}$ that are contained in ${U}$. (Thus, for instance, ${G}$ is NSS if and only if ${Q(U)}$ is trivial for all sufficiently small ${U}$.)

Proposition 7 (From metrisable to subgroup trapping) Let ${G}$ be a locally compact metrisable group. Then ${G}$ has the subgroup trapping property: for every open neighbourhood ${U}$ of the identity, there exists another open neighbourhood ${V}$ of the identity such that ${Q(V)}$ generates a subgroup ${\langle Q(V) \rangle}$ contained in ${U}$.

Proposition 8 (From subgroup trapping to NSS) Let ${G}$ be a locally compact group with the subgroup trapping property, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is locally compact and NSS.

Proposition 9 (From NSS to the escape property) Let ${G}$ be a locally compact NSS group. Then there exists a left-invariant metric ${d}$ on ${G}$ generating the topology on ${G}$ which obeys the escape property (1) for some constant ${C}$.

Proposition 10 (From escape to the commutator estimate) Let ${G}$ be a locally compact group with a left-invariant metric ${d}$ that obeys the escape property (1). Then ${d}$ also obeys the commutator property (2).

It is clear that Propositions 6, 7, and 8 combine to give Theorem 4, and Propositions 9, 10 combine to give Theorem 5.

Propositions 610 are all proven separately, but their proofs share some common strategies and ideas. The first main idea is to construct metrics on a locally compact group ${G}$ by starting with a suitable “bump function” ${\phi \in C_c(G)}$ (i.e. a continuous, compactly supported function from ${G}$ to ${{\bf R}}$) and pulling back the metric structure on ${C_c(G)}$ by using the translation action ${\tau_g \phi(x) := \phi(g^{-1} x)}$, thus creating a (semi-)metric

$\displaystyle d_\phi( g, h ) := \| \tau_g \phi - \tau_h \phi \|_{C_c(G)} := \sup_{x \in G} |\phi(g^{-1} x) - \phi(h^{-1} x)|. \ \ \ \ \ (3)$

One easily verifies that this is indeed a (semi-)metric (in that it is non-negative, symmetric, and obeys the triangle inequality); it is also left-invariant, and so we have ${d_\phi(g,h) = \|g^{-1} h \|_\phi = \| h^{-1} g \|_\phi}$, where

$\displaystyle \| g \|_\phi = d_\phi(g,\hbox{id}) = \| \partial_g \phi \|_{C_c(G)}$

where ${\partial_g}$ is the difference operator ${\partial_g = 1 - \tau_g}$,

$\displaystyle \partial_g \phi(x) = \phi(x) - \phi(g^{-1} x).$

This construction was already seen in the proof of the Birkhoff-Kakutani theorem, which is the main tool used to establish Proposition 6. For the other propositions, the idea is to choose a bump function ${\phi}$ that is “smooth” enough that it creates a metric with good properties such as the commutator estimate (2). Roughly speaking, to get a bound of the form (2), one needs ${\phi}$ to have “${C^{1,1}}$ regularity” with respect to the “right” smooth structure on ${G}$ By ${C^{1,1}}$ regularity, we mean here something like a bound of the form

$\displaystyle \| \partial_g \partial_h \phi \|_{C_c(G)} \ll \|g\|_\phi \|h\|_\phi \ \ \ \ \ (4)$

for all ${g,h \in G}$. Here we use the usual asymptotic notation, writing ${X \ll Y}$ or ${X=O(Y)}$ if ${X \leq CY}$ for some constant ${C}$ (which can vary from line to line).

The following lemma illustrates how ${C^{1,1}}$ regularity can be used to build Gleason metrics.

Lemma 11 Suppose that ${\phi \in C_c(G)}$ obeys (4). Then the (semi-)metric ${d_\phi}$ (and associated (semi-)norm ${\|\|_\phi}$) obey the escape property (1) and the commutator property (2).

Proof: We begin with the commutator property (2). Observe the identity

$\displaystyle \tau_{[g,h]} = \tau_{hg}^{-1} \tau_{gh}$

whence

$\displaystyle \partial_{[g,h]} = \tau_{hg}^{-1} ( \tau_{hg} - \tau_{gh} )$

$\displaystyle = \tau_{hg}^{-1} ( \partial_h \partial_g - \partial_g \partial_h ).$

From the triangle inequality (and translation-invariance of the ${C_c(G)}$ norm) we thus see that (2) follows from (4). Similarly, to obtain the escape property (1), observe the telescoping identity

$\displaystyle \partial_{g^n} = n \partial_g + \sum_{i=0}^{n-1} \partial_g \partial_{g^i}$

for any ${g \in G}$ and natural number ${n}$, and thus by the triangle inequality

$\displaystyle \| g^n \|_\phi = n \| g \|_\phi + O( \sum_{i=0}^{n-1} \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} ). \ \ \ \ \ (5)$

But from (4) (and the triangle inequality) we have

$\displaystyle \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} \ll \|g\|_\phi \|g^i \|_\phi \ll i \|g\|_\phi^2$

and thus we have the “Taylor expansion”

$\displaystyle \|g^n\|_\phi = n \|g\|_\phi + O( n^2 \|g\|_\phi^2 )$

which gives (1). $\Box$

It remains to obtain ${\phi}$ that have the desired ${C^{1,1}}$ regularity property. In order to get such regular bump functions, we will use the trick of convolving together two lower regularity bump functions (such as two functions with “${C^{0,1}}$ regularity” in some sense to be determined later). In order to perform this convolution, we will use the fundamental tool of (left-invariant) Haar measure ${\mu}$ on the locally compact group ${G}$. Here we exploit the basic fact that the convolution

$\displaystyle f_1 * f_2(x) := \int_G f_1(y) f_2(y^{-1} x)\ d\mu(y) \ \ \ \ \ (6)$

of two functions ${f_1,f_2 \in C_c(G)}$ tends to be smoother than either of the two factors ${f_1,f_2}$. This is easiest to see in the abelian case, since in this case we can distribute derivatives according to the law

$\displaystyle \partial_g (f_1 * f_2) = (\partial_g f_1) * f_2 = f_1 * (\partial_g f_2),$

which suggests that the order of “differentiability” of ${f_1*f_2}$ should be the sum of the orders of ${f_1}$ and ${f_2}$ separately.

These ideas are already sufficient to establish Proposition 10 directly, and also Proposition 9 when comined with an additional bootstrap argument. The proofs of Proposition 7 and Proposition 8 use similar techniques, but is more difficult due to the potential presence of small subgroups, which require an application of the Peter-Weyl theorem to properly control. Both of these theorems will be proven below the fold, thus (when combined with the preceding posts) completing the proof of Theorem 1.

The presentation here is based on some unpublished notes of van den Dries and Goldbring on Hilbert’s fifth problem. I am indebted to Emmanuel Breuillard, Ben Green, and Tom Sanders for many discussions related to these arguments.