Automorphisms of Finite Groups pp 69-116 | Cite as

# 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.