# Which finite groups have low-degree essential cohomology?

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 $$\mathrm{H}^2_{\mathrm{gp}}(-; \mathbb{F}_2)$$?

• @ChrisGerig Awesome. And your Proposition 2.2 already answers all questions about $\mathbb{F}_2$-cohomology (or rather it moves them to ring theoretic questions, which in any specific low degree I can compute directly). Aug 23, 2022 at 18:03

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.

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: $$Ess(G)=\bigcap\lbrace(x)\;|\;x\in H^1(G;\mathbb Z/2\mathbb Z)\rbrace$$

• @Theo You can reach out to me if you're interested in related concepts: Back then I was thinking about what I call "virtually essential cohomology" (where you ignore maximal subgroups) and formed a filtration on group cohomology, but neither me nor Ken Brown found a use for it. Aug 23, 2022 at 19:49