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 ↵ 2 Aut(G) such that ↵(g) = h. The fusion class of g 2 G is denoted by F(g) and defined as {↵(g)|↵ 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 ↵ of G is called a class-preserving automorphism if for each x 2 G, there exists an element gx 2 G such that ↵(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 ↵(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 di↵erent order. We also find PA(G) for a finite p-group having a cyclic maximal subgroup. As a consequence of our results, we deduce that if G is a finite p-group having a cyclic maximal subgroup, then |G| divides | Aut(G)|. In Chapter 3, we find PA(G) when G is a direct product of m cyclic p-groups of di↵erent orders. As a consequence of our results, we find PA(G) for any finite abelian group G. In Chapter 4, we prove that if G is a finite non-cyclic group, then PA(G)  5/8. We classify all finite groups with PA(G) 5/8. We prove that for any finite abelian group G, PA(G)=3/4 if and only if G ' C4, and PA(G)=2/3 if and only if G is isomorphic to C3 or C6. We also prove that there does not exist any finite group with PA(G)=5/8. At the end, we show that PA(G) = P(G) if and only if Aut(G) = Autc(G). This gives another perspective on a question of Avinoam Mann [31, Question 10]. The probability that two elements of G, selected randomly and chosen independently, are conjugate is denoted by (G). Let Pa(G) denote the probability that two elements of G, selected randomly and with replacement, are fused. In Chapter 5, we find Pa(G) for some finite groups, and calculate some upper and lower bounds of Pa(G). We prove that there does not exist any group G with Pa(G)|G| = 7/4. We prove if G is a finite abelian group such that each non-autocentral fusion class is of order 2, then G ' C2, C3, C4 or C2 ⇥ C3. We also prove that Pa(G) = (G) if and only if Aut(G) = Autc(G).