Implications of permutizers of some subgroups in finite groups. (English) Zbl 1075.20010
Let \(H\) be a subgroup of a finite group \(G\). We say that the subgroup \(P_G(H)=\langle x\mid\langle x\rangle H=H\langle x\rangle\rangle\) is the permutizer of \(H\) in \(G\). If every proper subgroup of \(G\) is properly contained in its permutizer, then \(G\) is said to satisfy the permutizer condition.
Many results have been obtained by using the permutizer condition. We mention the papers by J. C. Beidleman and D. J. S. Robinson [J. Algebra 191, No. 2, 686-703 (1997; Zbl 0880.20015)] and by R. Esteban-Romero and the reviewer [Commun. Algebra 30, No. 12, 5757-5770 (2002; Zbl 1018.20013)]. Beidleman and Robinson proved that every chief factor of \(G\) has order \(4\) or a prime if \(G\) satisfies the permutizer condition and Esteban-Romero and the reviewer showed that the converse is not true.
In the paper under review, the authors still get the same results as Beidleman and Robinson by using a weaker condition: \(M<P_G(M)\) if \(M\) is a maximal subgroup of \(G\) or \([G:M]\) is a power of a prime number. The authors also use permutizers of maximal subgroups to characterize supersolubility.
Many results have been obtained by using the permutizer condition. We mention the papers by J. C. Beidleman and D. J. S. Robinson [J. Algebra 191, No. 2, 686-703 (1997; Zbl 0880.20015)] and by R. Esteban-Romero and the reviewer [Commun. Algebra 30, No. 12, 5757-5770 (2002; Zbl 1018.20013)]. Beidleman and Robinson proved that every chief factor of \(G\) has order \(4\) or a prime if \(G\) satisfies the permutizer condition and Esteban-Romero and the reviewer showed that the converse is not true.
In the paper under review, the authors still get the same results as Beidleman and Robinson by using a weaker condition: \(M<P_G(M)\) if \(M\) is a maximal subgroup of \(G\) or \([G:M]\) is a power of a prime number. The authors also use permutizers of maximal subgroups to characterize supersolubility.
Reviewer: Adolfo Ballester-Bolinches (Burjasot)
MSC:
| 20D40 | Products of subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20E28 | Maximal subgroups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
References:
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| [2] | Baumeister B., Arch. Math. (Basel) 3 pp 167– (1999) · Zbl 0931.20018 · doi:10.1007/s000130050318 |
| [3] | Beidleman J. C., J. Algebra 191 pp 686– · Zbl 0880.20015 · doi:10.1006/jabr.1996.6932 |
| [4] | Guo X. Y., J. Math (PRC) 9 (2) (1989) |
| [5] | DOI: 10.1007/978-3-642-64981-3 · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3 |
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