# 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 where both and 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:

where is the exterior square of and also coincides with the Schur multiplier of .

Note that since locally cyclic implies epabelian, the Schur multiplier of is the trivial group, and all extensions are themselves abelian groups. In particular, , and we get:

Thus, the computation of is equivalent to the computation of .