Orders of Automorphism Groups of Finite Groups

Abstract

The object of study in this chapter is the relation between the order of a finite group and that of its group of automorphisms. In 1954, Scott [114] conjectured that a finite group has at least a prescribed number of automorphisms if the order of the group is sufficiently large. The conjecture was confirmed by Ledermann and Neumann [80, Theorem 6.6] in 1956 by constructing an explicit function $f: \mathbb {N} \rightarrow \mathbb {N}$ with the property that if the finite group G has order $|G |\ge f(n)$, then $|{\text {Aut}} (G)| \ge n$. In the same year, building on the techniques from [80], the authors [81] proved the following local version of Scott’s conjecture: Conjecture 3.1. There exists a function $f:\mathbb {N} \rightarrow \mathbb {N}$ such that for each $h \in \mathbb {N}$ and each prime p, if G is any finite group such that $p^{f(h)}$ divides |G|, then $p^h$ divides $|{\text {Aut}} (G)|$. Later on, Green [49], Howarth [63] and Hyde [68] successively improved the function f to a quadratic polynomial function. The aim of this chapter is to give an exposition of these developments. Schur multiplier plays a significant role in these investigations.