# zbMATH — the first resource for mathematics

$$C$$-normality of groups and its properties. (English) Zbl 0847.20010
A subgroup $$H$$ of finite group $$G$$ is called $$c$$-normal in $$G$$ if there exists a normal subgroup $$N$$ of $$G$$ such that $$G=HN$$ and $$H\cap N\leq H_G$$, the normal core of $$H$$ in $$G$$. The author derives several results relating the structure of $$G$$ and its $$c$$-normal maximal subgroups. Theorem 3.1. $$G$$ is solvable if and only if every maximal subgroup of $$G$$ is $$c$$-normal in $$G$$. Theorem 3.2. A maximal subgroup $$M$$ of $$G$$ is $$c$$-normal in $$G$$ if and only if $$\eta(G:M)$$, the normal index of $$M$$ in $$G$$, $$=[G:M]$$. Theorem 4.1. $$G$$ is supersolvable if $$P_1$$ is $$c$$-normal in $$G$$ for every maximal subgroup $$P_1$$ of each Sylow subgroup $$P$$ of $$G$$.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D25 Special subgroups (Frattini, Fitting, etc.) 20D35 Subnormal subgroups of abstract finite groups 20D40 Products of subgroups of abstract finite groups 20E28 Maximal subgroups
Full Text: