On the Probability that an Automorphism of a Group Fixes an Element of the Group

Abstract

Let G be a finite group and let Aut(G) denote the full automorphism group of G. An element a 2 G is said to be a conjugate of another element b 2 G if there exists a g 2 G such that a = g1bg. The relation of conjugacy on a group G is an equivalence relation and it partitions G into equivalence classes. The equivalence class or conjugacy class of an element b 2 G is denoted by Cl(b) and is defined as Cl(b) = {a 2 G | there exists g 2 G with a = g1bg}. Two elements g and h of G are said to be fused in G if there exists an automorphism and#8629; 2 Aut(G) such that and#8629;(g) = h. The fusion class of g 2 G is denoted by F(g) and defined as {and#8629;(g)|and#8629; 2 Aut(G)}. Let L(G) denote the set of all those elements of G which are fixed by all automorphisms of G. The set L(G) is called the autocentre of G. A fusion class is called an autocentral fusion class if it is contained in the autocenter, and a non-autocentral fusion class otherwise. An automorphism and#8629; of G is called a class-preserving automorphism if for each x 2 G, there exists an element gx 2 G such that and#8629;(x) = g1 x xgx; and is called an inner automorphism if for all x 2 G, there exists a fixed element g 2 G such that and#8629;(x) = g1xg. The groups of all classpreserving automorphisms and inner automorphisms of G are denoted by Autc(G) and Inn(G) respectively. In this thesis, Chapter 1 contains the introduction and some basic definitions. Let P(G) denote the probability that two randomly selected elements of G commute, and let PA(G) denote the probability that a randomly chosen automorphism of a finite group G fixes a randomly chosen element of G. In Chapter 2, we classify all finite abelian groups G such that PA(G)=1/p, where p is the smallest prime dividing | Aut(G)|. We classify all finite abelian groups G such that PA(G)=1/p, where p is any prime. We find PA(G) when G is a direct product of three or four cyclic p-groups of diand#8629;erent order. We also find PA(G) for a finite p-group having a cyclic maximal subgroup.