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**On \(c\)-supplemented and cover-avoidance properties of finite groups.**
*(English)*
Zbl 1173.20014

The paper under review presents some necessary and sufficient conditions for a finite group to be soluble, \(p\)-nilpotent, and supersoluble in terms of \(c\)-supplementation and the cover-avoidance properties of maximal subgroups of Sylow subgroups.

Recall that a subgroup \(H\) of a group \(G\) is \(c\)-supplemented in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\leq H_G\), the core of \(H\) in \(G\). This concept was introduced by A. Ballester-Bolinches, Y. Wang, and X. Guo [Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]. Recall also that a subgroup \(H\) is a CAP-subgroup of a group \(G\) or satisfies the cover and avoidance property in \(G\) when \(H\) either covers or avoids any chief factor of any chief series of \(G\).

The authors characterise soluble groups as the groups in which every subgroup is either \(c\)-supplemented or a CAP-subgroup (Theorem 2.4). For groups \(G\) with a normal subgroup \(H\) such that \(G/H\) is \(p\)-nilpotent, where \(p\) is a prime divisor of \(G\) such that \(|G|\) and \(p-1\) are coprime, they show that if every maximal of a Sylow \(p\)-subgroup of \(H\) is either \(c\)-supplemented in \(G\) or a CAP-subgroup of \(G\), then \(G\) is \(p\)-nilpotent (Theorem 3.1). For supersoluble groups, they show that if \(G\) has a normal subgroup \(H\) such that \(G/H\) is supersoluble and every maximal subgroup of any Sylow subgroup of \(H\) is either \(c\)-supplemented in \(G\) or a CAP-subgroup of \(G\), then \(G\) is supersoluble (Theorem 4.3). This result is generalised for saturated formations containing the class of all supersoluble groups by enforcing the condition on the maximal subgroups of the Sylow subgroups of the Fitting subgroup of \(H\) (Theorem 4.8).

Recall that a subgroup \(H\) of a group \(G\) is \(c\)-supplemented in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\leq H_G\), the core of \(H\) in \(G\). This concept was introduced by A. Ballester-Bolinches, Y. Wang, and X. Guo [Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]. Recall also that a subgroup \(H\) is a CAP-subgroup of a group \(G\) or satisfies the cover and avoidance property in \(G\) when \(H\) either covers or avoids any chief factor of any chief series of \(G\).

The authors characterise soluble groups as the groups in which every subgroup is either \(c\)-supplemented or a CAP-subgroup (Theorem 2.4). For groups \(G\) with a normal subgroup \(H\) such that \(G/H\) is \(p\)-nilpotent, where \(p\) is a prime divisor of \(G\) such that \(|G|\) and \(p-1\) are coprime, they show that if every maximal of a Sylow \(p\)-subgroup of \(H\) is either \(c\)-supplemented in \(G\) or a CAP-subgroup of \(G\), then \(G\) is \(p\)-nilpotent (Theorem 3.1). For supersoluble groups, they show that if \(G\) has a normal subgroup \(H\) such that \(G/H\) is supersoluble and every maximal subgroup of any Sylow subgroup of \(H\) is either \(c\)-supplemented in \(G\) or a CAP-subgroup of \(G\), then \(G\) is supersoluble (Theorem 4.3). This result is generalised for saturated formations containing the class of all supersoluble groups by enforcing the condition on the maximal subgroups of the Sylow subgroups of the Fitting subgroup of \(H\) (Theorem 4.8).

Reviewer: Ramon Esteban-Romero (Valencia)

### MSC:

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D40 | Products of subgroups of abstract finite groups |

20D15 | Finite nilpotent groups, \(p\)-groups |

20D25 | Special subgroups (Frattini, Fitting, etc.) |