On Sylowizers in finite \(p\)-soluble groups. (English) Zbl 1142.20005
In a finite group \(G\) the Sylowizer of a \(p\)-subgroup \(R\) is a subgroup maximal with respect to having \(R\) a Sylow \(p\)-subgroup. This concept is due to W. Gaschütz [Math. Z. 122, 319-320 (1971; Zbl 0234.20008)], where the question of conjugacy of the Sylowizers for \(p\)-solvable groups \(G\) was studied. A. V. Borovik and E. I. Khukhro [Math. Zametki 19, 401-418 (1976; Zbl 0347.20011)] found examples of Sylowizers of different orders.
In this paper sufficient conditions are given for the conjugacy of the Sylowizers, such as, (1) \(p=2\) and the \(2\)-Sylow subgroups of the group \(G\) are either dihedral or quaternion; (2) the \(p\)-Sylow subgroups of the group \(G\) are of order at most \(p^3\); (3) \(p\) is an odd prime, the subgroup \(R\) is Abelian of maximal order.
In this paper sufficient conditions are given for the conjugacy of the Sylowizers, such as, (1) \(p=2\) and the \(2\)-Sylow subgroups of the group \(G\) are either dihedral or quaternion; (2) the \(p\)-Sylow subgroups of the group \(G\) are of order at most \(p^3\); (3) \(p\) is an odd prime, the subgroup \(R\) is Abelian of maximal order.
Reviewer: János Kurdics (Nyíregyháza)
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
References:
| [1] | Borovik, A.V., Khukhro, E.I.: Sylowizers of subgroups in solvable groups. In: Materials of the XIII All-Soviet Scientific Students’ Conference (Mathematics), Novosibirsk, pp. 6–7 (1975) |
| [2] | Borovik A.V. and Khukhro E.I. (1976). Groups of automorphisms of finite p-groups. Math. Zametki 19(3): 401–418 · Zbl 0347.20011 |
| [3] | Gaschütz W. (1971). Sylowisatoren. Math. Z. 122: 319–320 · Zbl 0234.20008 · doi:10.1007/BF01110166 |
| [4] | Gorenstein D. (1980). Finite Groups. Chelsea Publishing Company, New York · Zbl 0463.20012 |
| [5] | Huppert B. (1967). Endliche Gruppen I. Springer, Heidelberg · Zbl 0217.07201 |
| [6] | Huppert B. and Blackburn N. (1982). Finite Groups III. Springer, Heidelberg · Zbl 0514.20002 |
| [7] | Lausch H. (1975). On p-complements of Sylowizers. Math. Z. 142: 25 · doi:10.1007/BF01214845 |
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