On groups well represented as automorphism groups of groups

TL;DR

This paper characterizes groups that can be realized as automorphism groups of arbitrarily large groups, under the assumption of G"odel's axiom of constructibility, highlighting the importance of the group's center size.

Contribution

It provides a new characterization of groups as automorphism groups of large groups, assuming the axiom of constructibility, with a focus on the size of the center.

Findings

Existence of arbitrarily large groups with automorphism group isomorphic to a given group L.

A single such group H with a sufficiently large center suffices for the characterization.

The characterization depends on the assumption V=L (constructibility).

Abstract

Assuming G\"{o}del's axiom of constructibility $\bold V=\bold L,$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H) \cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $ 2^{|L |+\aleph_0}$.