Second cohomology group up to isoclinism

Definition

Suppose $G$ is a group and $A$ is an abelian group. The second cohomology group up to isoclinism is the quotient of the usual second cohomology group by the following equivalence relations: two extensions $E_1,E_2$ are isoclinic as extensions if there is an isoclinism between them that is compatible with the identity maps for $G$ and $A$.

Relation with formula for second cohomology group

Consider the case that the action is trivial, i.e., we are looking at the second cohomology group for trivial group action.

Then, the second cohomology group up to isoclinism can be identified as the group of homomorphisms:

$\mathrm{H}\mathrm{o}\mathrm{m}(H_2(G;\mathbb{Z}),A)$

where $H_2(G;\mathbb{Z})=M(G)$ is the Schur multiplier.

Further, the second cohomology group up to isoclinism can be viewed as the quotient part of the short exact sequence arising from the formula for second cohomology group for trivial group action in terms of second homology group and abelianization:

$0\to {\mathrm{E}}_{\mathrm{x}}^{\mathrm{t}}(G^{\mathrm{a}\mathrm{b}},A)\to H^2(G;A)\to \mathrm{H}\mathrm{o}\mathrm{m}(H_2(G;\mathbb{Z}),A)\to 0$

In other words, it is the group $\mathrm{H}\mathrm{o}\mathrm{m}(H_2(G;\mathbb{Z}),A)$. Note that $H_2(G;\mathbb{Z})$ is the Schur multiplier, so this is in fact the group of homomorphisms from the Schur multiplier to $A$.

Related notions

• Second cohomology group up to isologism is a generalization from the variety of abelian groups.
• Formula for second cohomology group for trivial group action in terms of Baer invariant and verbal factor group