Mathematics and Statistics Vol. 10(6), pp. 1340 - 1343
DOI: 10.13189/ms.2022.100621
Reprint (PDF) (307Kb)


Central Automorphisms in n-abelian Groups


Rugare Kwashira *
Faculty of Science, School of Mathematics, University of the Witwatersrand, South Africa

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 Autc(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 Autc(G) and Autc∗ (G) where Autc∗ (G) is the group of central automorphisms that fix Z(G) point-wise. For instance, [6], if Autc(G) = Aut(G) then G is nilpotent of class 2 and if G is f-nilpotent for Autc∗ (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 Autc(G) and Autc∗ (G) are trivial, but for the case when G is a p-group, Autc(G) is non-trivial and the structure of Autc∗ (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 Autc∗ (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 = Gl that fix the center elementwise and consequently the algebraic structure of Autc∗ (G). For this goal, we will study the invariant normal subgroups M of G such that 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 Gl. We extend our study to central automorphisms in this class of groups Gl, in which the mapping is an automorphism. For such groups, Autc∗ (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 and Autc∗ (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.

KEYWORDS
Central Automorphism, n-abelian, Pullback, Group Action, Maximal Subgroup

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Rugare Kwashira , "Central Automorphisms in n-abelian Groups," Mathematics and Statistics, Vol. 10, No. 6, pp. 1340 - 1343, 2022. DOI: 10.13189/ms.2022.100621.

(b). APA Format:
Rugare Kwashira (2022). Central Automorphisms in n-abelian Groups. Mathematics and Statistics, 10(6), 1340 - 1343. DOI: 10.13189/ms.2022.100621.