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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.

A subgroup H of a group G is c-subnormal in G if G has a subnormal subgroup T such that HT=G and T 3 H ⊆ HG.^[1] Using this concept, in Jaraden obtain^[1] some new conditions for solubility of a finite group are given. Here we obtain local versions of these results

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

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