## ON THE ORBITS OF SOME METABELIAN GROUPS

## ON THE ORBITS OF SOME METABELIAN GROUPS

*K. Sharma, V. K. Bhat*

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## Abstract

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 consid- ered in this study are some dihedral groups D13, D14 and D15 as well as the semidirect products D7 β Z2, D5 β Z3 and D3 β Z5. 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 βxes 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.