Conjugacy classes in finite solvable groups (Q789507)

In this note the number of conjugacy classes in a finite solvable group is obtained as a function of any type of the composition factors of G. It results in a new elementary proof of one of Mann's results for solvable groups and that result is improved here for some classes of groups. \textit{A. Mann}'s result [Isr. J. Math. 31, 78-84 (1978)] is: \(| G| \equiv r(G)(mod ab),\) where \(| G| =p_ 1\!^{e_ 1}...p_ t\!^{e_ t}\), \(p_ i\) prime and \(p_ i\neq p_ j\) for every \(i\neq j\), \(r(G)=number\) of conjugacy classes of G, \(a=g.c.d.(p_ 1-1,...,p_ t-1), b=g.c.d.(p_ 1\!^ 2-1,...,p_ t\!^ 2-1).\)