SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS.

Suppose that G is a finite solvable group. Let t = nc(G) denote the number of orders of nonnormal subgroups of G. We bound the derived length dl(G) in terms of nc(G). If G is a finitep-group, we show that |G'| ≤ p2t+1 and dl(G) ≤ |"log2(2t + 3)]. If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of |G'| is less than t and that dl(G) ≤ ⌊2(t + 1)/3⌋ + 1. [ABSTRACT FROM AUTHOR]