New lower bounds for the number of conjugacy classes in finite nilpotent groups

P. Hall’s classical equality for the number of conjugacy classes in p-groups yields k(G) ≥ (3/2) log2 |G| when G is nilpotent. Using only Hall’s theorem, this is the best one can do when |G| = 2^n . Using a result of G.J. Sherman, we improve the constant 3/2 to 5/3, which is best possible across all nilpotent groups and to 15/8 when G is nilpotent and |G| 6= 8, 16. These results are then used to prove that k(G) log3 (|G|) when G/N is nilpotent, under natural conditions on N E G. Also, when G 0 is nilpotent of class c, we prove that k(G) ≥ (log |G|)^t when |G| is large enough, depending only on (c, t).