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.
Highlights
Classifying finite nilpotent groups is a difficult task, though there are some nilpotent groups that have been classified
The number of conjugacy classes was found for finite nilpotent groups in Lopez (1985), it is obtained as a function of the orders of certain subgroups
Zapirain (2011) established a lower bound for the number of conjugacy classes of finite nilpotent groups using the group size multiply by a certain constant c
Summary
Classifying finite nilpotent groups is a difficult task, though there are some nilpotent groups that have been classified. We will denote clG as the number of conjugacy classes of a group G. The number of conjugacy classes was found for finite nilpotent groups in Lopez (1985), it is obtained as a function of the orders of certain subgroups. Zapirain (2011) established a lower bound for the number of conjugacy classes of finite nilpotent groups using the group size multiply by a certain constant c. Let L = c(|G|1/c-1)+1 be the lower bound obtained by Sherman (c is the nilpotency class of G), L ≤ clG ≤ |G| (the trivial upper bound |G| will diverge for large size groups). All of the previous research obtained lower bounds for the value of clG or obtained this value for certain groups and not for all nonabelian nilpotent groups.
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