I discussed the binary polyhedral groups in the second post on the ADE diagrams. A more general class of groups has cropped up a couple of times recently, namely finite groups containing a unique involution. I’d like to discuss them here, since there is a nice cohomological argument due to George Glauberman which is not as well known as it should be.

Suppose that *G* is a finite group with a unique involution (element of order 2), *z*. Let *Z* be the subgroup generated by *z*, a normal (indeed, central) subgroup of *G* of order 2; and let *H* be the quotient *G*/*Z*.

Since *G* has a unique involution, the same is true of any subgroup of *G* of even order, in particular, any Sylow 2-subgroup of *G*. Now the 2-groups with unique involution were determined by Burnside; they must be cyclic or generalized quaternion groups. Now if *P* is a Sylow 2-subgroup of *G*, then *P*/*Z* is a Sylow 2-subgroup of *H*, and is cyclic or dihedral.

Now the natural way to proceed is to determine the groups with cyclic or dihedral Sylow 2-subgroups, and investigate which such groups *H* can be “lifted” to groups *G* with unique involution. The first step follows from two of the major theorems of “pre-classification” finite group theory, namely the Feit–Thompson and Gorenstein–Walter theorems. The second step can then be done with quite a lot of work.

But for many purposes, a simpler result suffices; this has a beautiful short proof.

Let *H* be a finite group with cyclic or dihedral Sylow 2-subgroups. Then there is a group *G* with a unique involution *z* such that *G*/*Z* is isomorphic to *H* (where *Z* is generated by *z*); and *G* is uniquely determined by *H*, up to isomorphism.

For example, the symmetric group *S*_{4} has dihedral Sylow 2-subgroups; there are two groups *G* with a central subgroup *Z* of order 2 such that *G*/*Z* is isomorphic to *S*_{4}. One of these is GL(2,3), which has many involutions; the other is the binary octahedral group, which has only one.

Here is the proof. Let *Q* be a Sylow 2-subgroup of *H* (a cyclic or dihedral group). It is straightforward that there is a unique extension *P* of *C*_{2} by *Q* which has a unique involution (so is cyclic or generalized quaternion). In fact, extensions of *C*_{2} by *Q* correspond to elements of the cohomology group H^{2}(*Q*,*C*_{2}); the uniqeness mentioned is actually the fact that there is a unique element of the cohomology group corresponding to such an element.

Let *t* be a cohomology class of a subgroup *S* of *G*. Then, for any *g* in *G*, there is a corresponding cohomology class *t ^{g}* of the conjugate

*S*. The class

^{g}*t*is said to be

*stable*if the images of

*t*and

*t*under the restriction maps from

^{g}*S*and

*S*to their intersection are equal, for any element

^{g}*g*of

*G*. The strong uniqueness above shows that our cohomology class for

*Q*is indeed stable.

A theorem of Cartan and Eilenberg shows that, if *t* is a stable cohomology class of a subgroup *S*, then the image of *t* under the composition of the corestriction (induction) map from *S* to *G* and the restriction map from *G* to *S* is simply *t* multiplied by the index of *S* in *G*. In our case, 2*t*=0, and the index of *Q* in *H* is odd, so *t* is fixed by this composite map. This means that, if *u* is the element of H^{2}(*H*,*C*_{2}) which is the image of *t* under the corestriction map, then the restriction map takes *u* to *t*. So the extension of *C*_{2} by *H* corresponding to the class *u* has the property that its Sylow subgroups are cyclic or generalized quaternion, so has a unique involution.

The proof also shows that *u* is unique with this property, which is a little stronger than saying that *G* is the unique extension of *H* with a unique involution (up to isomorphism).

Thus cohomology, which is not too far removed from “abstract nonsense” in some people’s view of things, has effortlessly performed a construction which would be a very great deal of effort to do directly!

Laci Babai and I used this result to characterise the abstract groups which are the full automorphism groups of switching classes of tournaments as the groups with cyclic or dihedral Sylow 2-subgroups. There were a couple of notable things about our paper (in the Electronic Journal of Combinatorics): we used random methods to prove the existence of objects with specified automorphism group, perhaps the first time this had been done; and the paper had a long and tangled history, which I have mentioned before on this blog.