The commutator subgroup made abelian

Abstract:

A theorem on covering spaces is proved which yields the following information about a group $\pi$, its commutator subgroup $\pi ’$ and their abelianizations: If ${\pi ^{ab}} \cong {Z_{{p^n}}}$, a cyclic group of order a power of the prime $p$, then $\pi {’^{ab}} = p\pi {’^{ab}}$. Hence if $\pi$ is also finitely generated, then $\pi {’^{ab}}$ is finite of order prime to $p$.