Abstract
Abstract
Suppose that G is a finite solvable group. Let
$t=n_c(G)$
denote the number of orders of nonnormal subgroups of G. We bound the derived length
$dl(G)$
in terms of
$n_c(G)$
. If G is a finite p-group, we show that
$|G'|\leq p^{2t+1}$
and
$dl(G)\leq \lceil \log _2(2t+3)\rceil $
. 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
$dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$
.
Publisher
Cambridge University Press (CUP)