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On the normal index and the $$c$$-section of maximal subgroups of a finite group. (English) Zbl 1203.20028
Let $$G$$ a finite group and $$M$$ a maximal subgroup of $$G$$; the normal index of $$M$$ in $$G$$, denoted by $$\eta(G:M)$$, is the order of a chief factor $$H/K$$ such that $$H$$ is a minimal supplement of $$M$$ in $$G$$. If $$H/K$$ is such a chief factor then $$G=MH$$, $$K\leq M$$ and $$(M\cap H)/K$$ is called a $$c$$-section of $$M$$. Finally if $$p\in\pi(G)$$ then $$\mathcal F_p(G)$$ is the set of maximal subgroups $$M$$ of $$G$$ such that $$(|G:M|,p)=1$$ and $$\mathcal F^p(G)$$ is the set of maximal subgroups $$M$$ of $$G$$ such that $$N_G(P)\leq M$$, where $$P\in\mathrm{Syl}_p(G)$$.
Among other results the authors prove the following: Theorem 3.1. A group $$G$$ is solvable if and only if $$\eta(G:M)_2=1$$ for every $$M\in\mathcal F^2(G)$$.
Theorem 3.4. A group $$G$$ is solvable if and only if a $$c$$-section of $$M$$ is either a $$2'$$-group or an Abelian $$2$$-group for every $$M\in\mathcal F^2(G)$$.
Theorem 3.8. Let $$G$$ be a group and $$p\in\pi(G)$$. Then $$G$$ is solvable if and only if $$\eta(G:M)$$ is a power of a prime for every $$M\in\mathcal F_p(G)$$.

##### MSC:
 20E28 Maximal subgroups 20D40 Products of subgroups of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups
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