Automorphisms of groups and converse of Schur's theorem

TL;DR

This paper classifies certain finitely generated nilpotent groups of class 2 based on their automorphism groups and provides simple proofs of the converse of Schur's theorem and related results.

Contribution

It offers a complete classification of finitely generated nilpotent groups of class 2 where IA-automorphisms are inner, and simplifies proofs of classical theorems.

Findings

Classification of finitely generated nilpotent groups of class 2 with IA(G) ≅ Inn(G)

Characterization of finite nilpotent groups where all IA-automorphisms are inner

Simplified proofs of the converse of Schur's theorem

Abstract

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results.