## 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)].

### 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

### Citations:

Zbl 0847.20010; Zbl 0929.20015
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