Lifting Automorphisms of Quotients by Central Subgroups

TL;DR

This paper investigates when automorphisms of quotients by central subgroups can be lifted to the original group, revealing a matrix equation approach and showing that inner automorphisms are not characteristic in automorphisms for certain metacyclic groups.

Contribution

It introduces a matrix equation method for lifting automorphisms in groups with central subgroups and demonstrates that inner automorphisms are not characteristic in automorphism groups of specific metacyclic groups.

Findings

Lifting automorphisms reduces to solving a matrix equation when N is central.

Inner automorphisms are not characteristic in Aut(G) for certain metacyclic groups.

Provides conditions under which automorphisms of quotients can be lifted.

Abstract

Given a finitely presented group $G$, we wish to explore the conditions under which automorphisms of quotients $G/N$ can be lifted to automorphisms of $G$. We discover that in the case where $N$ is a central subgroup of $G$, the question of lifting can be reduced to solving a certain matrix equation. We then use the techniques developed to show that $Inn(G)$ is not characteristic in $Aut(G)$, where $G$ is a metacyclic group of order $p^n$, $p\neq 2$.