Automorphisms of Finite Groups pp 117-155 | Cite as

# Groups with Divisibility Property-I

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## Abstract

Every finite non-cyclic abelian of the automorphism group \({\text {Aut}} (G)\) of

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

*G*.## Copyright information

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