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While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$, and $\chi(N)$, if defined (for $H^{i}(H,\mathbb{F}_{p}), H^{i}(G, \mathbb{F}_{p}), H^{i}(N, \mathbb{F}_{p})$), satisfy the relation $\chi(G) = \chi(H) \cdot \chi(N)$. The paper referred to the book Galois cohomology of Serre, and I found it in an exercise (exercise (c) in page 29 I.§ 4.1).

I find this statement surprising and have not found a reference yet. Is there some simple explanation for this fact?

If there is some reference for the proof of this fact (in group cohomology) or a more general cohomology theory, it will help me a lot.

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    $\begingroup$ For general $G$ virtually torsion-free this is Brown, Chapter IX, Proposition 7.3(d) (page 250). $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Apr 26, 2023 at 18:53
  • $\begingroup$ @Carl-FredrikNybergBrodda Thank you very much for your helpful answer. I will study that part! $\endgroup$
    – gualterio
    Commented Apr 27, 2023 at 4:32

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