Groups of prime-power order with a small second derived quotient

Abstract

For odd primes we prove some structure theorems for finite $p$-groups $G$, such that $G^{\prime \prime }\ne 1$ and $|G^{\prime }/G^{\prime \prime }|=p^3$. Building on results of Blackburn and Hall, it is shown that $\lcs G3$ is a maximal subgroup of $G^{\prime }$, the group $G$ has a central decomposition into two simpler subgroups, and, moreover, $G^{\prime }$ has one of two isomorphism types.

16 pages