On a finite \(\pi\)-solvable group with a supersolvable \(\pi\)-Hall subgroup

D. Gritsuk,   V. Monakhov

All groups considered in this paper are finite. Each \(\pi\)-solvable group \(G\) has a subnormal series \[G=G_0\supseteq G_1\supseteq \ldots \supseteq G_{n-1}\supseteq G_n=1 \] whose factors \(G_{i-1}/G_{i}\) are \(\pi^{\prime}\)-groups or abelian (or nilpotent) \(\pi\)-group. The least number of abelian (nilpotent) \(\pi\)-factors of all such subnormal series of \(G\) is called the derived (respectively, nilpotent) \(\pi\)-length of \(G\) and denoted by \(l_{\pi}^a(G)\) (respectively, by \(l_{\pi}^n(G)\)). Clearly, \(l_{\pi}^n(G)\le l_{\pi}^a(G)\) for any \(\pi\)-solvable group \(G\). Some estimates of these \(\pi\)-lengths are established in [1–4]. In particular, if \(G\) is a \(\pi\)-solvable group in which the derived subgroup of a \(\pi\)-Hall subgroup is nilpotent, then \[l_{\pi}^n(G) \leq 1+ \max_{r \in \pi} l_r(G),\] see [1]. We obtained an analogue of these results for the derived \(\pi\)-length.

Theorem 1. Let \(G\) be a \(\pi\)-solvable group. If the derived subgroup of a \(\pi\)-Hall subgroup of \(G\) is nilpotent, then \(l_{\pi}^a(G) \leq 1 + \max _{r \in \pi} l_r^a(G).\)

Since the derived subgroup of a supersolvable group is nilpotent, Theorem 1 implies

Corollary 1. Let \(G\) be a \(\pi\)-solvable group. If a \(\pi\)-Hall subgroup of \(G\) is supersolvable, then \(l_{\pi}^a(G) \leq 1+\max_{r\in \pi}l_r^a(G).\)

Corollary 2. Let \(G\) be a \(\pi\)-solvable group. If a Sylow \(p\)-subgroup of \(G\) is cyclic for every \(p \in \pi\), then \(l_{\pi}^a(G) \leq 2\).

Corollary 3. Let \(G\) be a \(\pi\)-solvable group, and let a Sylow \(p\)-subgroup of \(G\) be bicyclic for every \(p \in \pi\). If \(2 \notin \pi\), then \(l_{\pi}^a(G) \leq 3.\)

References

1. V. S. Monakhov, O. A. Shpyrko, On nilpotent \(\pi\)-length of a finite \(\pi\)-solvable group, Discrete Mathematics., 13, N3 (2001), 145–152. (in Russian)
2. D. V. Gritsuk, V. S. Monakhov , O. A. Spyrko, On derived \(\pi\)-length of a \(\pi\)-solvable group, BSU Vestnik, Series 1, N3 (2012), 90–95. (in Russian)
3. D. V. Gritsuk, V. S. Monakhov, O. A. Spyrko, On finite \(\pi\)-solvable groups with bicyclic Sylow subgroups, Problems in Physics, Mathematics, and Technics, N14 (2013), 61–66. (in Russian)
4. D. V. Gritsuk, V. S. Monakhov, On solvable groups whose Sylow subgroups are either abelian or extra special, Proc. Inst. Math. NAS Belarus 20, N2 (2012), 3–9. (in Russian)

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