Automorphism Groups of Finite p-Groups

Abstract

Let G be a group and let Aut(G) denote the group of all automorphisms of G. 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 x element g 2 G such that (x) g1xg. 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 ' of G is called a central automorphism if it commutes with all inner automorphisms of G; or equivalently g1'(g) 2 Z(G), the center of G, for all g 2 G. The group of all central automorphisms of G is denoted as Autz(G). The main objective in this thesis is to study the equality of two automorphism groups of a finite p-group. In chapter 2, we give necessary and sufficient conditions for a fnite p-group G such that Aut(G) Autz(G). We classify all nite p-groups G such that Autc(G) Autz(G) when Z(G) is cyclic or elementary abelian, or when order of the group is at most p7. We complete the classification of all nite p-groups of order p5 for which there exist non-inner class-preserving automorphisms. In chapter 3, we study _nite p-groups G for which Autz(G) is of minimal order, that is, Autz(G) = Z(Inn(G)). We give necessary and sufficient conditions on a nite p group G of coclass upto 4 such that Autz(G) Z(Inn(G)), and as a consequence, we characterize all nite pgroups G of order at most p7 for which Autz(G) Z(Inn(G)). Our results generalize the results obtained by Sharma and Gumber [57]. In 1904, Schur [56] proved that if the index of the center of a group is nite, then its commutator subgroup is also nite. In chapter 4, we give an elementary and different proof of the converse of this theorem using automorphisms. Our results generalize the previous results obtained in this direction by Niroomand [46] and Sury [58]. For a nite group G, the structures of Inn(G); Aut(G) and Autz(G) are inuenced by the structure of the conjugacy classes of G In the nal chapter, we study nite groups in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) Union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) Union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G