Central Automorphisms in n-abelian Groups

Abstract

The study of Aut(G), the group of automorphisms of G, has been undertaken by various authors. One way to facilitate this study is to investigate the structure of Aut_c(G), the subgroup of central automorphisms. For some classes of groups, algebraic properties like solvability, nilpotency, abelian and nilpotency relative to an automorphism can be deduced through the study of the subgroups Aut_c(G) and Aut_c∗ (G) where Aut_c∗ (G) is the group of central automorphisms that fix Z(G) point-wise. For instance, [6], if Aut_c(G) = Aut(G) then G is nilpotent of class 2 and if G is f-nilpotent for <img src=image/13428941_01.gif> Aut_c∗ (G), then for a group G, the notions of relative nilpotency and nilpotency coincide [8]. The group is abelian if G is identity nilpotent only [8]. For an arbitrary group G, the subgroups Aut_c(G) and Aut_c∗ (G) are trivial, but for the case when G is a p-group, Aut_c(G) is non-trivial and the structure of Aut_c∗ (G) have been described [4]. The study of the influence of types of subgroups on the structure of G is a powerful technique, thus, one can investigate the influence of maximal invariant subgroups of G on the structure of Aut_c∗ (G). We shall consider a class of finite, non-commutative, n-abelian groups that are not necessarily pgroups. Here, n = 2l + 1 is a positive integer and l is an odd integer. The purpose of this paper is to explicitly describe the central automorphisms of G = G_l that fix the center elementwise and consequently the algebraic structure of Aut_c∗ (G). For this goal, we will study the invariant normal subgroups M of G such that <img src=image/13428941_02.gif> and M is maximal in G. It suffices to study Hom(G/M,Z(G)), the group of homomorphisms from the quotient G/M to the center Z(G). We explore the central automorphism group of pullbacks involving groups of the form G_l. We extend our study to central automorphisms in this class of groups G_l, in which the mapping <img src=image/13428941_03.gif> is an automorphism. For such groups, Aut_c∗ (G) can be described through Hom(G/M,Z(G)), where M is normal and a maximal subgroup in G such that the quotient group G/M is abelian. We show that Hom<img src=image/13428941_04.gif> and Aut_c∗ (G) is isomorphic to the cyclic group of order a prime p. The class of groups studied in our paper falls under a bigger class of groups which have a special characterization that their non normal subgroups are contranormal. The results of this paper can be generalized to this bigger class of groups.