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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 $H\cap K\le {\text{core}}_{G}\left(H\right)$, 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 $\left(|G|,p-1\right)=1$. 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 $G/{O}_{p}\left(G\right)$ is $p$-nilpotent and every ${p}^{\text{'}}$-subgroup of $G$ is contained in some Hall ${p}^{\text{'}}$-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 $G/{O}_{p}\left(G\right)$ 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)].

MSC:
 20D40 Products of subgroups of finite groups 20D20 Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure 20D35 Subnormal subgroups of finite groups