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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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