Isomorphic subgroups of solvable groups. (English) Zbl 1330.20028
Let \(G\) be a finite group, and let \(H\) and \(K\) be subgroups of \(G\). Suppose that \(H\) is a maximal subgroup of \(G\), and \(H\) is isomorphic to \(K\). Then, it is not necessarily the case that \(K\) is also a maximal subgroup of \(G\). However, it is not known whether the hypotheses above imply that \(K\) is a maximal subgroup of \(G\) whenever \(G\) is solvable.
In this paper, the authors prove this implication holds in some cases. In particular, from their Theorem A it follows that the implication holds when \(G\) is solvable and \(H\) is supersolvable. In their Theorem B they show that the implication holds whenever \(G\) is solvable and \(H\) has an abelian Sylow \(2\)-subgroup. For the proof of Theorem B, they use some properties of nilpotent injectors and Glauberman’s ZJ-Theorem.
In this paper, the authors prove this implication holds in some cases. In particular, from their Theorem A it follows that the implication holds when \(G\) is solvable and \(H\) is supersolvable. In their Theorem B they show that the implication holds whenever \(G\) is solvable and \(H\) has an abelian Sylow \(2\)-subgroup. For the proof of Theorem B, they use some properties of nilpotent injectors and Glauberman’s ZJ-Theorem.
Reviewer: Alexandre Turull (Gainesville)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20E28 | Maximal subgroups |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
finite groups; solvable groups; nilpotent injectors; Glauberman ZJ-theorem; maximal subgroups; Sylow \(2\)-subgroups; supersolvable subgroupsReferences:
| [1] | Glauberman, George, A characteristic subgroup of a \(p\)-stable group, Canad. J. Math., 20, 1101-1135 (1968) · Zbl 0164.02202 |
| [2] | Lausch, Hans, Conjugacy classes of maximal nilpotent subgroups, Israel J. Math., 47, 1, 29-31 (1984) · Zbl 0535.20008 · doi:10.1007/BF02760560 |
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