Structure of the subgroup of a finite group which has trivial intersection with every maximal subgroup. (Russian) Zbl 0561.20018
Investigation of normal and subgroup structure of finite groups, Proc. Gomel’ Semin., Minsk 1984, 33-38 (1984).
[For the entire collection see Zbl 0551.00003.]
The author studies groups G with a subgroup H with the following property: if M is a maximal subgroup of G then \(H\leq M\) or \(H\cap M=1\). Obviously \(H\leq \Phi (G)\) or \(H\cap \Phi (G)=1\). If G is \(\pi\)-soluble then H has a normal p-complement for every \(p\in \pi\). For related results see S. Bauman [Ill. J. Math. 21, 568-574 (1977; reviewed below), Arch. Math. 25, 337-340 (1974; Zbl 0291.20028)].
The author studies groups G with a subgroup H with the following property: if M is a maximal subgroup of G then \(H\leq M\) or \(H\cap M=1\). Obviously \(H\leq \Phi (G)\) or \(H\cap \Phi (G)=1\). If G is \(\pi\)-soluble then H has a normal p-complement for every \(p\in \pi\). For related results see S. Bauman [Ill. J. Math. 21, 568-574 (1977; reviewed below), Arch. Math. 25, 337-340 (1974; Zbl 0291.20028)].
Reviewer: Ya.G.Berkovich
MSC:
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D40 | Products of subgroups of abstract finite groups |
