On the capability of groups

Abstract

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.1991 Mathematics subject classification: 20F29.

On the nonabelian tensor square and capability of groups of order p 2 q

“…(see [9,Proposition 1]) Let G be a finitely generated capable group. Then every central element z in G has order dividing exp ((G/<z>) ab ).…”

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

“…It can be easily shown that Z ⊗ (G) and Z ∧ (G) are characteristic and central subgroups of G with Z ⊗ (G) ⊆ Z ∧ (G). With the help of the exterior centre, Ellis in [9] obtained the desired external characterisation of the epicentre as follows. THEOREM 1.6 ([9]).…”

“…Thus, G satisfies the assumptions of Theorem 3.1 and it follows that a group of Type (10) is not capable. Now let G be a group of Type (9). We observe that G/G is free abelian of rank 2 and thus satisfies the necessary condition for capability as given in Theorem 3.2.…”

“…As in all cases before, this still requires the cumbersome process of evaluating factor sets in its implementation. Ellis [9] was able to characterise the epicentre with the help of the non-abelian tensor square, and thus made the determination of the epicentre computationally accessible. The non-abelian tensor square is defined as follows.…”

Computing the nonabelian tensor squares of polycyclic groups

“…The tensor center of H can be characterized as the largest subgroup A of H such that(H/A) ⊗ (H/A) ∼ = H ⊗ H[15]. The exterior center is equal to Z * (H), the epicenter of H[12]. Groups with trivial epicenters are called capable[2].…”