An Upper Bound to the Number of Conjugacy Classes of Non-Abelian Nilpotent Groups

Abstract

Abstract:The number of conjugacy classes of symmetric group, dihedral group and some nilpotent groups is obtained. Until now, it has not been obtained for all nilpotent groups. Although there are some lower bounds to this value, there is no non-trivial upper bound. This paper aims to investigate an upper bound to this number for all finite nilpotent groups. Moreover, the exact number of conjugacy classes is found for a certain case of non-abelian nilpotent groups.

ON NON-ABELIAN GROUPS OF ORDER 2^n, n >= 4 USING GAP

“…The conjugacy class of x ∈ G is the set of all elements in G that conjugate to x, and denoted by Cl G (x). The conjugation performs a partition for the group G, see [1]. So, if…”