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$$C$$-normality and solvability of groups. (English) Zbl 0853.20015
A subgroup $$H$$ of group $$G$$ is $$c$$-normal in $$G$$ if there exists a normal subgroup $$N$$ of $$G$$ such that $$HN = G$$ and $$H \cap N$$ lies in the core of $$H$$. Theorem 3.1. A finite group $$G$$ is solvable if and only if every maximal subgroup of $$G$$ is $$c$$-normal in $$G$$. Theorem 3.2. A finite group $$G$$ is solvable if and only if $$G$$ contains a solvable $$c$$-normal maximal subgroup. A condition for $$p$$-solvability is also obtained.

MSC:
 20D40 Products of subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20E28 Maximal subgroups 20D25 Special subgroups (Frattini, Fitting, etc.)
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References:
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