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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.