Which finite groups have low-degree essential cohomology?

Question

Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with coefficients in $A$. Recall that a class $\alpha \in \mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ is essential if it is nonzero, but the restriction $\alpha|_S \in \mathrm{H}^\bullet_{\mathrm{gp}}(S; A)$ is zero for all proper subgroups $S \subsetneq G$. For example, a standard lemma shows that $G$ has no essential cohomology if $G$ is not a $p$-group for some prime $p$ (as the collection of Sylow $p$-subgroups together detect all cohomology). On the other hand, for a cyclic group of prime order, all classes are essential, just because the group has no subgroups.

General question: Which groups have essential cohomology of low degree?

The question is basically trivial in degree $1$. Indeed, $\mathrm{H}^1_{\mathrm{gp}}(G; A) = \hom(G, A)$, and a nonzero homomorphism takes a nonzero value on some element, so the only groups with essential cohomology in degree $1$ are cyclic (and these do have essential cohomology). Moreover, it's approximately true that an abelian group has essential cohomology of degree $d$ when it has rank $d$; compare Which groups have undetectable third U(1)-cohomology?.

Specific question: Which groups have essential classes in $\mathrm{H}^2_{\mathrm{gp}}(-; \mathbb{F}_2)$?

Answer 1

Call a finite $p$ group $G$ $p$--central if every element of order $p$ is central.

In a 1997 Comm. Math. Helv. paper, A. Adem and D. Karagueuzian proved that every $p$--central group has essential mod $p$ cohomology. I got at this theorem in a different way in [N.J.Kuhn, Adv. Math. 216 (2007), 387--442], with a result that defines a number $e(G)$ that gives the degree of a particular essential element with other nice properties (e.g. it is acted on trivially by all Steenrod operations).

So what is this number $e(G)$? Let $C$ be the (unique) maximal elementary abelian subgroup of $G$: the nontrivial elements of $C$ are all the elements of $G$ of order $p$. Then $e(G)$ is determined by the image of the restriction $ H^*(G;\mathbb Z/p) \rightarrow H^*(C;\mathbb Z/p)$, which is always a sub-Hopf algebra of $H^*(C;\mathbb Z/p)$. More precisely, $e(G)$ is the top degree of the finite dimensional Hopf algebra $H^*(C;\mathbb Z/p) \otimes_{H^*(G;\mathbb Z/p)} \mathbb Z/p$, and is often quite easy to compute.

$e(G)$ is often quite easy to compute, and, in tables in an appendix of the paper, I list the values of $e(G)$ for all $2$--central $2$--groups of order up through 64. In particular, the following indecomposable 2-groups have $e(G)=2$, and thus essential classes in degree 2: groups of order 32 with Hall-Senior numbers 19, 21, 29, and 30, and groups of order 64 with Hall-Senior numbers 38, 39, 41, 64, 65, 140, and 141.

A general result would go: Proposition Let $G$ be a $2$--central group with $C$ of rank 2. If the image of the restriction map (as above) is polynomial on two classes in dimension $2$, then $e(G)=2$ and so there is essential cohomology in degree 2.

None of this is very trivial. Read my paper (and a later follow-up of mine on nilpotence in cohomology) if your are interested. The theory is worked out at odd primes too; I was just obsessed with $2$-groups as I looked for examples. I should also say that I was not looking for the essential cohomology class of lowest degree; indeed, my class is likely the essential class of greatest degree.

Answer 2

If I were to attempt a general p-group classification I'd use the theory of extensions (for degree 2). Regardless, I wrote a paper as an undergrad to compute the essential cohomology of a sub-classification (all p-groups with a cyclic subgroup of index p) from which you can read off what you want: https://arxiv.org/abs/1006.4836. Likewise a sub-classification for elementary abelian p-groups was given by Aksu-Green 2009, and for extraspecial p-groups by Minh 2000.

For p = 2 there is another classical result (due to Marx 1990 but probably Quillen knew it) that handles mod 2 essential cohomology in all degrees: $$Ess(G)=\bigcap\lbrace(x)\;|\;x\in H^1(G;\mathbb Z/2\mathbb Z)\rbrace$$