On central automorphisms of groups and nilpotent rings

TL;DR

This paper investigates the structure of the central automorphism group of a group using the concept of the adjoint group of a ring, providing new insights into its size and structure in general cases.

Contribution

It introduces a method leveraging the adjoint group of a ring to analyze the structure of central automorphisms of groups more generally.

Findings

Provides formulas for the size of $Aut_c(G)$

Analyzes the structure of $Aut_c(G)$ in special cases

Uses the notion of the adjoint group of a ring as a key tool

Abstract

Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ \sigma \mapsto h_\sigma$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_\sigma(x)=x^{-1} \sigma(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the main tool in our approach.