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Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$. Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...

Let $1\to A\to B\to C\to 1$ be an exact sequence of (nonabelian) $G$-groups. Then there is a well-known exact sequence of pointed sets $ 1\to A^G\to B^G\to C^G\to H^1(G,A)\to H^1(G,B)\to H^1(G,C) $ ...

Let $A$ be an abelian group, and $p$ a prime. I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$ If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...

Edit: According to the comments of abx and Yemon Choi I revise the question as follows: Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...