# Formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization

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## Formula

Suppose $G$ is an abelian group and $A$ is an abelian group. Then, the second cohomology group for trivial group action $H^2(G;A)$ occurs in the following natural short exact sequence (this sequence arises as the $n = 2$ case of the dual universal coefficients theorem for group cohomology):

$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$

Since $G$ is also an abelian group, we have $G^{\operatorname{ab}} = G$ and the Schur multiplier $H_2(G;\mathbb{Z})$ becomes the exterior square $\bigwedge^2G$ (see Schur multiplier of abelian group is its exterior square), and in this case, the short exact sequence becomes:

$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$

The sequence splits (though not necessarily naturally), so we get:

$H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \oplus \operatorname{Hom}(\bigwedge^2G,A)$

This can be interpreted as follows:

• The group $\operatorname{Ext}^1_{\mathbb{Z}}(G,A)$ describes the abelian groups arising as extensions with $G$ on top of $A$.
• The group $\operatorname{Hom}(\bigwedge^2G,A)$ can be viewed as group of all alternating $\mathbb{Z}$-bilinear maps from $G$ to $A$.

## Interpretation of left map of the sequence

The mapping sends each abelian group extension with $G$ on top of $A$ to the same extension now viewed as a cohomology class. The image of the mapping is the subgroup $H^2_{sym}(G,A)$ comprising the cohomology classes that contain symmetric 2-cocycles.

## Interpretation of right map of the sequence

• In cocycle terms: The mapping from $H^2(G;A)$ to $\operatorname{Hom}(\bigwedge^2G,A)$ sends a given cohomology class to the skew of any 2-cocycle representing it (the skew of $f:G \times G \to A$ is $(x,y) \mapsto f(x,y) - f(y,x)$).
• In group extension terms: The mapping sends an extension group $E$ to the commutator map of $E$, which can be viewed as an alternating bilinear map $G \times G \to A$.
• In isoclinism terms: The group $\operatorname{Hom}(\bigwedge^2G,A)$ can be viewed as the second cohomology group up to isoclinism for the trivial group action of $G$ on $A$. The mapping that we have can thus be viewed as taking an element of $H^2(G;A)$ to its equivalence class under isoclinism.

## Comment on direct sum

As an internal direct sum, the summand for $\operatorname{Ext}^1_{\mathbb{Z}}(G,A)$ is $H^2_{sym}(G;A)$. Because of the non-naturality of splitting, the other summand cannot usually be identified explicitly as a particular subgroup of $H^2(G;A)$.