Isomorphism between automorphism groups of finitely generated groups

TL;DR

This paper provides a simplified proof of the conditions under which the group of central automorphisms fixing the center elementwise is isomorphic to the inner automorphism group in finitely generated groups, also extending to new related results.

Contribution

The authors present a concise proof of existing conditions for automorphism group isomorphism and introduce new related results for finitely generated groups.

Findings

Simplified proof of Azhdari and Malayeri's conditions

New proofs of related automorphism group results

Extension of isomorphism conditions to additional cases

Abstract

Let $G$ be a finitely generated group and let $C^*$ denote the group of all central automorphisms of $G$ fixing the center of $G$ elementwise. Azhdari and Malayeri [J. Algebra Appl., {\bf 6}(2011), 1283-1290] gave necessary and sufficient conditions on $G$ such that $C^* \simeq \mathrm{Inn}(G)$. We prove a technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri. Subsequently, we also obtain short proofs of some other existing and some new related results.