Second cohomology group for trivial group action of locally cyclic torsion-free groups

The goal of this page is to discuss the second cohomology group for trivial group action $H^2(G;A)$ where both $G$ and $A$ are locally cyclic aperiodic groups.

Computation of the group

We use the formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization (which is a special case of the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization), namely the following short exact sequence: $0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$

where $\bigwedge^2G = M(G)$ is the exterior square of $G$ and also coincides with the Schur multiplier of $G$.

Note that since locally cyclic implies epabelian, the Schur multiplier of $G$ is the trivial group, and all extensions are themselves abelian groups. In particular, $\bigwedge^2G = 0$, and we get: $H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A)$

Thus, the computation of $H^2(G;A)$ is equivalent to the computation of $\operatorname{Ext}^1_{\mathbb{Z}}(G;A)$.