On the structure of automorphism groups of finite groups

Abstract

Let G be an arbitrary group and let Aut(G) denote the full automorphism group of G. An automorphism α of G is called a class-preserving automorphism if for each x ∈ G, there exists an element g ∈ G such that α(x) = g−1xg ; and is xxx called an inner automorphism if for all x ∈ G, there exists a fix element g ∈ G such that α(x) = g−1xg. The group Inn(G) of all inner automorphisms of G is a normal subgroup of the group Autc(G) of all class-preserving automorphisms of G. An automorphism α of G is called an nth class-preserving if for each x ∈ G, there exists an element gx ∈ γn(G), where γn(G) denotes the nth term of the lower central series of G, such that α(x) = g−1xg . The set Autn(G) of all nth class-preserving xxc automorphisms of G fixing Z(G) element-wise is a normal subgroup of Aut(G). An automorphism α of G is called a central automorphism if it commutes with all inner automorphisms of G; or equivalently g−1α(g) ∈ Z(G), the center of G, for all g ∈ G. The group of all central automorphisms of G is denoted as Autcent(G). An automorphism α of a group G is called a commuting automorphism if each element x in G commutes with its image α(x) under α . Let A(G) denote the set of all commuting automorphisms of G. Observe that Autcent(G) is contained in A(G). A group G is called an A(G)-group if the set A(G) is a subgroup of Aut(G). In this thesis, we mainly study the structure of A(G), Autcent(G) and Autnc (G). We find some conditions on a finite p-group G such that A(G) is a subgroup of Aut(G). We also find conditions on a finite p-group G such that Autnc (G) = Autcent(G) and Autcent(G) = Z(Inn(G)). Chapter 1, contains the introductory part and some basic definitions. In chapter 2, we give necessary and sufficient conditions for a finite p-group G of class n + 1 such that Autnc (G) = Autcent(G). As a consequence, we give a short proof of the main result of Yadav [45] which states that: Let G be a finite p-group of class 2 and let pm1 , . . . , pmd be the invariants of G/Z(G). Then Autc(G) = Autcent(G) if and only if γ2(G) = Z(G) and | Autc(G)| = 􏷌di=1 |Ωmi (γ2(G))|. A group G is said to be metacyclic if it contains a cyclic normal subgroup Z such v

vi that G/Z is cyclic. In chapter 3, we find some necessary and sufficient conditions on a finite non-abelian p-group G, where p is odd prime, with G/Z(G) metacyclic such that G is an A(G)-group. For x ∈ G, let [x,G] denote the set {[x,g]|g ∈ G}. A non-abelian group G that has no non-trivial abelian direct factor is said to be purely non-abelian. Let G be a finite p-group and N be a non-trivial normal subgroup of G. The pair (G,N) is called a Camina pair if N ⊆ [x,G] for all x ∈ G−N. A finite p-group G is called Frattinian if Z(M) ̸= Z(G) for all maximal subgroups M of G. A Frattinian p-group G satisfying CG(Z(Φ(G))) = Φ(G) is called strongly Frattinian. In chapter 4, we find some necessary conditions on a finite non-abelian p-group G such that G is an A(G)-group. In this chapter we give two theorems. In first one, we prove that if G is a finite non-abelian p-group such that CG(Φ(G)) is cyclic, then G is an A(G)-group. In second one, we prove that if G is a finite Frattinian p-group such that G/Z(G) is purely non-abelian and (G,Z(G)) is a Camina pair, then G is strongly Frattinian, and if p is odd, then G is an A(G)-group. In chapter 5, we study finite p-groups G for which Autcent(G) is of minimal order, that is, Autcent(G) = Z(Inn(G)). We give necessary and sufficient conditions on a finite p-group G such that Autcent(G) = Z(Inn(G)).