# The extension class of a finite Heisenberg group

Let $$\mathbb{K}$$ be a field of characteristic $$\neq 2$$ and let $$(V, \omega)$$ be a symplectic vector space. Then the Heisenberg group $$\mathsf{Heis}(V, \, \omega)$$ is the central extension of the additive group $$V$$ $$$$1 \to \mathbb{K} \to \mathsf{Heis}(V, \, \omega) \to V \to 1 \quad \quad (*)$$$$ given as follows: the underlying set of $$\mathsf{Heis}(V, \, \omega)$$ is $$V \times \mathbb{K}$$, endowed with the group law $$$$(v_1, \, t_1)\,(v_2, \, t_2) = \left(v_1+v_2, \, t_1+t_2 + \frac{1}{2} \omega(v_1, \, v_2)\right).$$$$

If $$\mathbb{K}= \mathbb{F}_p$$, then $$V$$ is an elementary abelian $$p$$-group and $$\mathsf{Heis}(V, \, \omega)$$ is a an extra-special finite $$p$$-group. Moreover, since the extension $$(*)$$ is central, it defines a structure of trivial $$V$$-module on $$\mathbb{F}_p$$.

On the other hand, the cohomology algebra $$H^*(V, \, \mathbb{F}_p)$$, when the action is trivial, can be described as follows, see the Introduction to [AG09]:

Theorem. Let $$V$$ be an elementary abelian $$p$$-group, and let $$\mathbb{F}_p$$ be endowed with the structure of trivial $$V$$-module. Then there is an isomorphism of graded algebras $$H^*(V, \, \mathbb{F}_p) \simeq \Lambda(V^{\vee}) \otimes_{\mathbb{F}_p} S(V^{\vee}),$$ where the exterior copy of the dual space is $$H^1(V, \mathbb{F}_p)$$ and the polynomial copy lives in $$H^2( V, \, \mathbb{F}_p)$$; specifically, the polynomial copy is the image of the exterior copy under the Bockstein boundary map $$\beta \colon H^1(V, \mathbb{F}_p) \to H^2( V, \, \mathbb{F}_p)$$.

Now it seems (at least to me) raisonable to state the following

Conjecture. The cohomology class of the extension $$(*)$$ corresponds, under the above identification of the cohomology algebra $$H^*(V, \, \mathbb{F}_p)$$, to the element $$\omega \otimes 1 \in H^2(V, \, \mathbb{F}_p)$$, where $$\omega \in \Lambda^2(V^{\vee})$$ represents the non-degenerate alternating form on $$V$$ via the natural duality $$\Lambda^2(V^{\vee}) \simeq \mathrm{Alt}^2(V)$$.

So here is my

Question. Is the above conjecture true?

I am by no means an expert in group cohomology theory, so I apologize in advance if the answer turns out to be trivial for the experts in the field. Every reference to the relevant literature will be highly appreciated.

References.

[AG09] F. A. Aksu, D. J. Green: Essential cohomology for elementary abelian $$p$$-groups, Journal of Pure and Applied Algebra 213, Issue 12 (2009), 2238-2243.

• When V is 2-dimensional, I think that this is contained in Proposition 9.1 of the paper: I. Efrat and J. Minac, "On the descending central sequence of absolute Galois groups", Amer. J. Math. 133 (2011), 1503-1532. Dec 11 '18 at 18:55
• This follows from carefully tracking the universal coefficient theorem for group cohomology. See, e.g., groupprops.subwiki.org/wiki/… Dec 12 '18 at 4:32