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 S4 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 S4. 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 C2 by Q which has a unique involution (so is cyclic or generalized quaternion). In fact, extensions of C2 by Q correspond to elements of the cohomology group H2(Q,C2); 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 tg of the conjugate Sg. The class t is said to be stable if the images of t and tg under the restriction maps from S and Sg to their intersection are equal, for any element 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, 2t=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 H2(H,C2) which is the image of t under the corestriction map, then the restriction map takes u to t. So the extension of C2 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.