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).