New Lower Bounds for the Number of Conjugacy Classes in Finite Nilpotent Groups

Abstract

P.Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) >= (3/2)log_2 |G|when G is nilpotent. Using only Hall's theorem, this is the best one can do when |G| = 2^n. Using aresult of G.J. Sherman, we improve the constant 3/2 to 5/3, which is best possible across all nilpotentgroups and to 15/8 when G is nilpotent and |G| is not equal to 8 or 16. These results are then used to prove that k(G) > log_3 |G| when G/N is nilpotent, under natural conditions on N (normal in) G. Also,when G' is nilpotent of class c, we prove that k(G) >= (log |G|)^t when |G| is large enough, dependingonly on (c,t).