Capability and homological functors of infinite two - generator groups of nilpotency class two

Abstract

A group is called capable if it is a central factor group. Baer characterized finitely generated abelian groups which are capable as those groups which have two or more factors of maximal order in their direct decomposition. The capability of groups have been determined for infinite metacyclic groups and for 2-generator p-group of nilpotency class two (p prime). The remaining case for capability of 2-generator group of nilpotency class two is the infinite case where the groups have been classified by Sarmin in 2002. Let R be the class of infinite 2-generator groups of nilpotency class two. Using their classification and non-abelian tensor squares, the capability of groups in R are determined. Brown and Loday in 1984 and 1987 introduced the nonabelian tensor square of a group to be a special case of the nonabelian tensor product which has its origin in algebraic K-theory as well as in homotopy theory. The homological functors have been determined for infinite metacyclic groups and non-abelian 2-generator p-groups of nilpotency class two. Therefore, the homological functors including the exterior square, the symmetric square and the Schur multiplier of groups in R will also be determined in this research.