On Automorphism Groups of Finitely Generated Groups

Abstract

Let G be a group and let Aut(G) denote the group of all automorphisms of G. An automorphism φ of G is called a central automorphism if it commutes with all inner automorphisms of G; or equivalently g^(-1) φ(g)∈Z(G), the center of G, for all g∈G. The group of all central automorphisms of G is denoted as Aut^z (G). An automorphism α of G is called a class-preserving automorphism if for each x∈G, there exists an element g_x∈G such that α(x)=g_x^(-1) xg_x and is called an inner automorphism if for all x∈G, there exists a fix element g∈G such that α(x)=g^(-1) xg. The group of all inner automorphisms of G is denoted by Inn(G) and by Aut_c (G), we denote the group of all class-preserving automorphisms of G. The main objective of the thesis is to study isomorphism between two automorphism groups of a finitely generated group. Chapter 1, contains the introductory part and some basic definitions. Following Bachmuth [10], we call an automorphism φ of G an IA-automorphism if it induces the identity automorphism on the abelianlized group G/G^'. In other words, if g^(-1) φ(g)∈G^', the commutator subgroup of G, for all g∈G. In chapter 2, we classify all finitely generated nilpotent groups G of class 2 for which IA(G)≅Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. Following Hegarty, we analogously call an automorphism α an absolute central automorphism if g^(-1) α(g)∈L(G) for all g∈G, where L(G) is the absolute center of G. Let Aut^l (G) denote the group of all absolute central automorphisms of G and let Aut_z^l (G) denote the group of all those absolute central automorphisms of G which fix Z(G) element-wise. In chapter 3, we obtain very short and easy proofs of main results of Moghaddam and Safa [49]. We also study finitely generated groups G with G^'≤L(G) such that Aut^l (G)≅Inn(G)0 and, as a consequence, obtain the necessary and sufficient conditions on a finite p-group G such that Aut^l (G)=Inn(G) Our results generalize the results obtained by Nasrabadi and Farimani [52, Main Theorem]. In chapter 4, we prove a technical lemma, Lemma 4.2.1, and as a consequence give a short and easy proof of main theorem of Azhdari and Malayeri [8, Theorem 0.1]. We also obtain short and alternate proofs of Corollary 2.1 of Azhdari and Malayeri [9], Propostion 1.11 and Theorem 2.2(i) of Azhdari [7]. Some other related results for finitely generated and finite p-groups are also obtained. 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. A group G is said be of co-class 2 if it is nilpotent group of class n-2. Rai [Proc. Japan Acad., Ser. A 91 (2015), no. 5,57-60] has given some sufficient conditions on a finite p-group G such that A(G) is a subgroup of Aut(G) and, as a consequence, has proved that in a finite p-group G of co-class 2, where p is an odd prime, A(G) is a subgroup of Aut(G). In chapter 5, we give very elementary and short proofs of main results of Rai and obtain some other related results.