# Group Cohomology Vs Profinite group Cohomology

What is the difference of the group cohomology and the profinite group cohomology? I think one reason is that in the profinite group situation, the G-module must be continuous. Does any other difference?

I also want to find an profinite group whose group cohomology is not the profinite group cohomology. If there is not such a group, we need not to define the profinite group cohomology.

• My impression is that in a suitable context, they are the same. That context is topological groups. To a topological group, one mat construct $EG\to BG$ in a variety of ways (I like the simplicial bar construction my self). Then regular group (co)homology would be the (co)homology of $BG$ when $G$ is given the discrete topology and profinite group (co)homology is the (co)homology of $BG$ when $G$ is profinite. – Baby Dragon May 1 '13 at 13:59
• What do you mean by "what is the difference"? – Bruno Joyal Nov 22 '13 at 4:40
• Thanks for answering this question, I have a simple example at 4.2.4 in the book: Gille P, Szamuely T. Central simple algebras and Galois cohomology[J]. 2006. – Strongart Sep 15 '18 at 14:00

If you take a prime $p$ and a non-commutative finitely generated free pro-$p$ group $F$, then the second continuous cohomology group with coefficients in $\mathbb Z/p$ is trivial $$H^2_{\sf cont}(F,\mathbb Z/p)=0$$ but the discrete cohomology group is uncountable (in particular, it is nontrivial) $$H^2_{\sf disc}(F,\mathbb Z/p)\ne 0.$$