## On complemented subgroups of finite groups.(English)Zbl 0929.20015

Let $$G$$ be a finite group. If $$N$$ is a normal subgroup of $$G$$ such that $$G/N$$ is supersoluble and every minimal subgroup of $$N$$ is complemented in $$G$$, then $$G$$ is supersoluble (Theorem 1). If all maximal subgroups of every Sylow subgroup of $$G$$ are complemented in $$G$$, then $$G$$ is supersoluble (Theorem 2). If $$p$$ is the smallest prime dividing the order of $$G$$, all 2-maximal subgroups of every Sylow $$p$$-subgroup of $$G$$ are complemented in $$G$$ and $$G$$ is $$A_4$$-free, then $$G$$ is $$p$$-nilpotent (Theorem 3). If $$G$$ has even order and $$K$$ is a maximal subgroup of a Sylow-2 subgroup of $$G$$ such that $$K$$ is complemented in $$G$$, then $$G$$ is not nonabelian simple and a composition factor of $$G$$ is either a cyclic group of prime order or $$B=\text{PSL}(2,q)$$ for some Mersenne prime $$q$$ such that a Hall $$2'$$-subgroup of $$B$$ is the normaliser of a Sylow $$q$$-subgroup of $$B$$ (Theorem 4).

### 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
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