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: N→ N with the property that if the finite group G has order | G| ≥ f(n), then | Aut (G) | ≥ 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: N→ N such that for each h∈ N and each prime p, if G is any finite group such that pf(h) divides |G|, then ph divides | 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.