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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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