ON THE ORBITS OF SOME METABELIAN GROUPS

In this study, the orbits of non abelian metabelian groups of order 26, 28 and 30 are found using conjugation action. The non abelian metabelian groups considered in this study are some dihedral groups [D.sub.13], [D.sub.14] and [D.sub.15] as well as the semidirect products [D.sub.7] * [Z.sub.2], [D...

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Bibliographic Details
Published in:TWMS journal of applied and engineering mathematics 2022-07, Vol.12 (3), p.799
Main Authors: Sharma, Karnika, Bhat, Vijay Kumar
Format: Article
Language:English
Subjects:
Abelian groups
Commutativity
Commuting
Conjugation
Graph coloring
Graph theory
Mathematical research
Mathematics
Orbits
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Summary:In this study, the orbits of non abelian metabelian groups of order 26, 28 and 30 are found using conjugation action. The non abelian metabelian groups considered in this study are some dihedral groups [D.sub.13], [D.sub.14] and [D.sub.15] as well as the semidirect products [D.sub.7] * [Z.sub.2], [D.sub.5] * [Z.sub.3] and [D.sub.3] * [Z.sub.5]. In order to obtain the orbit, the set of all pairs of commuting elements of the groups of size two is found. The results obtained from the orbits are then applied into an extension of commutativity degree which is the probability that an element of the groups fixes a set. Finally, the generalized conjugacy class graph is constructed and its chromatic number and clique number are also found. Keywords: Metabelian group, conjugacy class graph, complete graph, group action.
ISSN:2146-1147
2146-1147