On Inter Relationships of Different 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 and#945; of G is called a class-preserving automorphism if for each x and#8712; G, there exists an element g and#8712; G such that and#945;(x) = gand#8722;1xg ; and is xxx called an inner automorphism if for all x and#8712; G, there exists a fix element g and#8712; G such that and#945;(x) = gand#8722;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 and#945; of G is called an IA-automorphism if xand#8722;1and#945;(x) and#8712; Gand#8242; for all x and#8712; G. Let IA(G) denote the group of all IA-automorphisms of G and let CIA(G)(Z(G)) denote the group of all IA-automorphisms of G fixing the center Z(G) of G elementwise. An automorphism and#945; of G is called a central automorphism if it commutes with all inner automorphisms of G; or equivalently gand#8722;1and#945;(g) and#8712; Z(G), the center of G, for all g and#8712; G. The group of all central automorphisms of G is denoted as Autz(G). Following Hegarty [38], we analogously call an automorphism and#945; an absolute central automorphism if gand#8722;1and#945;(g) and#8712; L(G) for all g and#8712; G, where L(G) is the absolute center of G. Let Var(G) and CVar(G)(Z(G)) respectively denote the group of all absolute central automorphisms of G and absolute central automorphisms of G fixing the center Z(G) of G elementwise. An automorphism and#945; of a group G is called a commuting automorphism if each element x in G commutes with its image and#945;(x) under and#945;. Let A(G) denote the set of all commuting automorphisms of G. Observe that Autz(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 CVar(G)(Z(G)), Autz(G) and A(G). We find necessary and sufficient conditions on a finite non-abelian p-group G such that CIA(G)(Z(G)) = CVar(G)(Z(G)) and CVar(G)(Z(G)) = Inn(G). We also find necessary and sufficient conditions on a finite purely non-abelian p-group G such that Var(G) = Autz(G) and CVar(G)(Z(G)) = Autz(G).