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 ≥ 2, then k(G) > |G|^1/(2^d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G⁽ⁱ⁾: G⁽ⁱ⁺¹⁾] ₁ ^d−1 , where G⁽ⁱ⁾ is theith derived group, leads to a |G|^1/(2d−1) lower bound for k(G), from which we derive a |G|^c/log ₂log₂|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.