Infinite metacyclic groups and their non-abelian tensor squares

Abstract

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group.

The exponents of nonabelian tensor products of groups

“…Thus even if G ∧ G is a finite group, this does not necessarily imply that G is locally finite of finite exponent. Consider for instance infinite metacyclic groups that have been classified by Beuerle and Kappe [4]. Let G = a, b | a m = 1, [a, b] = a 1−r , where m > 0, r ≥ 0, (r, m) = 1, r ≡ 1mod m. It is shown in [4] that G ∧ G ∼ = C m , a finite group.…”

“…Consider for instance infinite metacyclic groups that have been classified by Beuerle and Kappe [4]. Let G = a, b | a m = 1, [a, b] = a 1−r , where m > 0, r ≥ 0, (r, m) = 1, r ≡ 1mod m. It is shown in [4] that G ∧ G ∼ = C m , a finite group. On the other hand, G ⊗ G is nonperiodic [4,Theorem 4.3].…”

“…Let G = a, b | a m = 1, [a, b] = a 1−r , where m > 0, r ≥ 0, (r, m) = 1, r ≡ 1mod m. It is shown in [4] that G ∧ G ∼ = C m , a finite group. On the other hand, G ⊗ G is nonperiodic [4,Theorem 4.3].…”

The Exterior Squares of Some Crystallographic Groups

“…Kappe et al in [7] investigated on two-generator two-groups of class two and computed the nonabelian tensor squares of these groups. The computations of the nonabelian tensor squares of infinite metacyclic groups and free nilpotent groups of finite rank have been discussed in [8] and [9] respectively. The exterior squares have been computed for finite p  groups of nilpotency class two, infinite nonabelian 2-generator groups of nilpotency class two and symmetric groups of order six in [10], [11] and [12] respectively.…”

On the Capability of Finitely Generated Non-Torsion Groups of Nilpotency Class 2

“…In [5], tensor methods were used for the first time to determine the infinite metacyclic groups that are capable. Explicit knowledge of the tensor square of twogenerator p-groups of class 2, p an odd prime, enabled the authors of [3], to investigate the capability of such groups.…”