Finite p-groups of class 2 as central extensions

2020/08/25 by Chen, Haimiao
#20D15 #20D45 #20J05 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2008.10970

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 H²(G,A), compute H²(G,A), and describe the actions of \rm End(G) and \rm End(A) on H²(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 p⁷ whose automorphism groups are abelian.

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