Finite groups with \(\mathbb{P} \)-subnormal Sylow subgroups. (English. Ukrainian original) Zbl 1506.20044
Ukr. Math. J. 72, No. 10, 1571-1578 (2021); translation from Ukr. Mat. Zh. 72, No. 10, 1365-1371 (2020).
Summary: Let \(\mathbb{P}\) be the set of all prime numbers. A subgroup \(H\) of a finite group \(G\) is called \(\mathbb{P}\)-subnormal if either \(H = G\) or there exists a chain of subgroups \(H = H_0 \leq H_1 \leq \dots \leq H_n = G\) such that \(\vert H_i : H_{i - 1} \vert \in \mathbb{P}\), \(1 \leq i \leq n\). We prove that any finite group with \(\mathbb{P} \)-subnormal Sylow \(p\)-subgroup of odd order is \(p\)-solvable and any group with \(\mathbb{P} \)-subnormal generalized Schmidt subgroups is metanilpotent.
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D35 | Subnormal subgroups of abstract finite groups |
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