Finite $p$-groups of class $2$ as central extensions

Abstract

Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and describe the actions of ${\rm End}(G)$ and ${\rm End}(A)$ on $H^2(G,A)$. These are used to provide an efficient criterion for lifting endomorphisms of $G$ to homomorphisms between two central extensions. Subsequently, we present two applications to illustrate the usefulness of this approach, in the case $p>2$. First, we recover the classification of two-generator $p$-groups of class $2$ up to isomorphism, and compute the order of the automorphism group for each isomorphism class. Second, we construct a family of nonabelian $p$-groups of order $p^7$ whose automorphism groups are abelian.