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**Finite groups with some subgroups of Sylow subgroups \(c\)-supplemented.**
*(English)*
Zbl 0953.20010

The author introduces a generalization of both “being complemented” and his concept of \(c\)-normality [J. Algebra 180, No. 3, 954-965 (1996; Zbl 0847.20010)] as follows: a subgroup \(H\) of a group \(G\) is said to be \(c\)-supplemented (in \(G\)) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\leq\text{core}_G(H)\), the largest normal subgroup of \(G\) contained in \(H\).

Theorem 3.1: Let \(G\) be a finite group and let \(P\) be a Sylow \(p\)-subgroup of \(G\) where \(p\) is a prime divisor of \(|G|\) with \((|G|,p-1)=1\). Suppose that every maximal subgroup of \(P\) is \(c\)-supplemented in \(G\) and any two complements of \(P\) in \(G\) are conjugate in \(G\). Then \(G/O_p(G)\) is \(p\)-nilpotent and every \(p'\)-subgroup of \(G\) is contained in some Hall \(p'\)-subgroup of \(G\). Theorem 3.3: Let \(G\) be a finite group and let \(N\) be a normal subgroup of \(G\) such that \(G/N\) is supersoluble. If every maximal subgroup of every Sylow subgroup of \(N\) is \(c\)-supplemented in \(G\), then \(G\) is supersoluble. Theorem 4.2: Let \(G\) be a finite group and let \(p\) be the smallest prime divisor of \(|G|\). If \(G\) is \(A_4\)-free and every second-maximal subgroup of a Sylow \(p\)-subgroup of \(G\) is \(c\)-normal in \(G\), then \(G/O_p(G)\) is \(p\)-nilpotent. The last two theorems generalize results by A. Ballester-Bolinches and X. Guo [Arch. Math. 72, No. 3, 161-166 (1999; Zbl 0929.20015)].

Theorem 3.1: Let \(G\) be a finite group and let \(P\) be a Sylow \(p\)-subgroup of \(G\) where \(p\) is a prime divisor of \(|G|\) with \((|G|,p-1)=1\). Suppose that every maximal subgroup of \(P\) is \(c\)-supplemented in \(G\) and any two complements of \(P\) in \(G\) are conjugate in \(G\). Then \(G/O_p(G)\) is \(p\)-nilpotent and every \(p'\)-subgroup of \(G\) is contained in some Hall \(p'\)-subgroup of \(G\). Theorem 3.3: Let \(G\) be a finite group and let \(N\) be a normal subgroup of \(G\) such that \(G/N\) is supersoluble. If every maximal subgroup of every Sylow subgroup of \(N\) is \(c\)-supplemented in \(G\), then \(G\) is supersoluble. Theorem 4.2: Let \(G\) be a finite group and let \(p\) be the smallest prime divisor of \(|G|\). If \(G\) is \(A_4\)-free and every second-maximal subgroup of a Sylow \(p\)-subgroup of \(G\) is \(c\)-normal in \(G\), then \(G/O_p(G)\) is \(p\)-nilpotent. The last two theorems generalize results by A. Ballester-Bolinches and X. Guo [Arch. Math. 72, No. 3, 161-166 (1999; Zbl 0929.20015)].

Reviewer: Hans Lausch (Clayton)

### MSC:

20D40 | Products of subgroups of abstract finite groups |

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

20D35 | Subnormal subgroups of abstract finite groups |