LOWER BOUNDS FOR THE NUMBER OF CONJUGACY CLASSES IN FINITE SOLVABLE-GROUPS

We prove first that if G is a finite solvable group of derived length d greater-than-or-equal-to 2, then k(G) > \G\1/(2d-1), Where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G(i):G(i+1)]1d-1, where G(i) is the ith derived group, leads to a \G\1/(2d-1) lower bound for k(G), from which we derive a \G\c/log2log2\G\ lower bound, independent of d(G). Finally, "almost logarithmic" lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.