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Remarks on two generalizations of normality of subgroups. (Chinese. English summary) Zbl 1109.20017

Summary: A subgroup \(H\) is said to be semi cover-avoiding in a group \(G\) if there is a chief series \(1=G_0<G_1<\cdots<G_l=G\) such that for every \(j=1,\cdots,l\), either \(H\) covers \(G_j/G_{j-1}\) or \(H\) avoids \(G_j/G_{j-1}\). This paper shows that semi cover-avoidence is suitable to cover both \(C\)-normality and the cover-avoidence property, and to characterize the solvability of groups by means of maximal subgroups or Sylow subgroups.

MSC:

20D30 Series and lattices of subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D35 Subnormal subgroups of abstract finite groups
20E28 Maximal subgroups
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