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SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 27 December 2023, pp. 121-128
- Print publication:
- August 2024
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- Article
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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 
$dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$.
