Baer invariants and extension theory for cyclic groups

This article describes Baer invariants corresponding to various subvarieties of the variety of groups and the corresponding extension theory.

It turns out that for any variety containing the subvariety of abelian groups, the Baer invariant of any cyclic group is the trivial group. This is because, if we express the cyclic group as a quotient of a free group $$F = \mathbb{Z}$$ by a normal subgroup $$R = n\mathbb{Z}$$ (allow $$n = 0$$), then the verbal subgroup $$V(F)$$ is trivial on account of $$F$$ being in the subvariety. Thus, the Baer invariant, given as:

$$\mathcal{V}M(G) = \frac{R \cap V(F)}{V^*(R,F)}$$

is also a trivial group.

In particular, this implies that all the nilpotent multipliers and solvable multipliers are trivial groups.