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 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 IA-automorphism if x−1α(x) ∈ G′ for all x ∈ 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 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 Autz(G). Following Hegarty [38], we analogously call an automorphism α an absolute central automorphism if g−1α(g) ∈ L(G) for all g ∈ 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 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 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). We obtain some conditions on a finite non-abelian p-group G such that A(G) is a subgroup of Aut(G).

Chapter 1 contains the introductory part and some basic definitions. In chapter 2, we give necessary and sufficient conditions for a finite non-abelian p-group G such that CIA(G)(Z(G)) = CVar(G)(Z(G)). We also give necessary and sufficient conditions for a finite non-abelian p-group G such that CVar(G)(Z(G)) = Inn(G). Finally, we give three Gap algorithms. Algorithm 1 can be used to find the absolute center of a group G. Algorithm 2 checks whether an automorphism α of a group G is absolute central or not. Algorithm 3 can be used to find the size of all absolute central automorphisms of G. Notice that Var(G) is a normal subgroup of Autz(G) and if L(G) = Z(G), then Var(G) = Autz(G). A natural question which arises here is that if L(G) < Z(G), then under what conditions Var(G) = Autz(G)? In chapter 3, we give necessary and sufficient conditions for a finite purely non-abelian p-group G such that Var(G) = Autz(G). We also obtain necessary and sufficient conditions for a finite purely non- abelian p-group G such that CVar(G)(Z(G)) = Autz(G). We give an example of a purely non-abelian group which satisfies the hypothesis of our first theorem. Let AutΦ(G) denote the group of all automorphisms α of G such that x−1α(x) ∈ Φ(G), the Frattini subgroup of G, for all x ∈ G. In [58], Muller using cohomological methods proved that if G is a finite p-group which is neither elementary abelian nor extraspecial, then AutΦ(G)/Inn(G) is a nontrivial normal p-subgroup of the group of outer(non-inner) automorphisms of G; or equivalently, if AutΦ(G) = Inn(G), then G is either elementary abelian or extraspecial. In chapter 4, we obtain an alternate proof of the main result of Muller [58]. 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 5, we study finite p-groups G for which G is an A(G)-group and prove three theorems. In the first theorem, we prove that if G is a finite non- abelian p-group, p an odd prime, such that IA(G) = Inn(G) and |G/G′| = p2, then G is an A(G)-group. In the second theorem, we prove that if G is a finite non-abelian p-group of coclass 3, where p is an odd prime, and G is strongly Frattinian, then G is an A(G)-group. In the third theorem, we prove that if G be a finite non-abelian p-group and M1, M2 are any two distinct maximal abelain subgroups of G such that M1 ∩ M2 = Z(G), then G is an A(G)-group.