Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2

Abstract

The classification of infinite 2-generator groups of nilpotency class 2, up to isomorphism are given as follows: Theorem 1 Let be a 2-generator group of nilpotency class less than or equal to 2 of the form ⋊ , where is an infinite cyclic group and is a p-group. Then G is isomorphic to exactly one group of the following types: (1.1) ⋊ ,where (1.2) ⋊ , where (1.3) ⋊ ,where (1.4) where The groups in the above list have nilpotency class two precisely for (1.1), (1.2), and (1.3) and are abelian for (1.4). Theorem 2 Let G be an infinite non-abelian 2-generator group of nilpotency class two. Then G is isomorphic to exactly one group of the following types: (2.1) ⋊ where (2.2) ⋊ , where, for , the component is a -group, for and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively. Let R be the class of infinite 2-generator groups of nilpotency class 2 of Type 2.2. Using their classification and nonabelian tensor squares, the homological functors in R such as the exterior square, the symmetric square and the Schur multiplier are determined in the Composition Theorem as follows: Theorem 3 (Composotion Theorem). Let G be a group of Type 2.2, that is ⋊ , where, for , the componenst are -groups, pi a prime with and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively, then , (3.1) , (3.2) , (3.3) , (3.4) , (3.5) , (3.6) , (3.7) , (3.8) where T(H) is the torsion subgroup of a group H and Tp(H) denotes the p-torsion subgroup of a group H.