Groups with Divisibility Property-I

Abstract

Every finite non-cyclic abelian p-group of order greater than $p^2$ has the property that its order divides that of its group of automorphisms (Theorem 3.34). The problem whether every non-abelian p-group of order greater than $p^2$ possesses the same property has been a subject of intensive investigation. As discussed in the introduction, this property is referred to as the Divisibility Property. While several classes of p-groups have been shown to have Divisibility Property, it is now known that not all finite p-groups admit this property [46]. An exposition of these developments is presented in the remaining part of this monograph. In this chapter, some reduction results, due to Buckley [14], are presented in Sect. 4.1. Among other results, it is proved that one can confine attention to the class of purely non-abelian p-groups. In subsequent sections it is shown that Divisibility Property is satisfied by p-groups of nilpotency class 2 [33], p-groups with metacyclic central quotient [18], modular p-group [22], p-abelian p-groups [19], and groups with small central quotient [20]. In view of Theorem 3.34, it can be assumed that the groups under consideration are non-abelian p-groups. The main ingredient in verifying Divisibility Property for various classes of groups G is the subgroup

$${\text {IC}} (G):={\text {Inn}} (G){\text {Autcent}} (G)$$

of the automorphism group ${\text {Aut}} (G)$ of G.