A contribution to the theory of finite supersolvable groups. (English) Zbl 0399.20026
The relationship between nilpotence of finite groups and the normality of (i) all maximal subgroups or (ii) all Sylow subgroups, is well known. Using Sylow systems of a subgroup, the author defines concepts of weak normaliser and weak centraliser and uses these to describe properties of supersolvable groups. For example, previous results of the author [J. Algebra 44, 160–168 (1977; Zbl 0344.20016), Proc. Am. Math. Soc. 68, 140–142 (1978; Zbl 0376.20016)] show that: A finite group \(G\) is supersolvable if and only if either i) \(G\) has a weakly normal Sylow system or ii) each maximal subgroup of \(G\) is weakly normal or iii) each proper subgroup of \(G\) is weakly subnormal.
A weak central series of a group may also be defined as well as an analogue of the Fitting subgroup. In both cases, the author demonstrates that their relation to supersolvable groups is very similar to that of the original concepts to nilpotent groups. Finally, an analogue of the Burnside theorem which gives a condition for \(p\)-nilpotence is obtained.
A weak central series of a group may also be defined as well as an analogue of the Fitting subgroup. In both cases, the author demonstrates that their relation to supersolvable groups is very similar to that of the original concepts to nilpotent groups. Finally, an analogue of the Burnside theorem which gives a condition for \(p\)-nilpotence is obtained.
Reviewer: J. R. J. Groves (Parkville)
MSC:
| 20D30 | Series and lattices of subgroups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
References:
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| [2] | Glauberman, G., Factorizations in Local Subgroups of Finite Groups, (CBMS No. 33 (1977), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R.I.) · Zbl 0489.20012 |
| [3] | Huppert, B., Zur Theorie der Formationen, Arch. Math. (Basel), 19, 561-574 (1968) · Zbl 0192.35303 |
| [4] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0217.07201 |
| [5] | Ito, N., Remarks on factorizable groups, Acta. Sci. Math. (Szeged), 14, 83-84 (1951) · Zbl 0044.01503 |
| [6] | Kegel, O., On Hupperts characterization of finite supersoluble groups, (Proc. Internat. Conf. Theory of Groups. Proc. Internat. Conf. Theory of Groups, Canberra, 1965 (1967), Gordon & Breach: Gordon & Breach New York), 209-215 · Zbl 0178.02101 |
| [7] | Venzke, P., System quasinormalizers in finite solvable groups, J. Algebra, 44, 160-168 (1977) · Zbl 0344.20016 |
| [8] | Venzke, P., Maximal subgroups of prime index in a finite solvable group, (Proc. Amer. Math. Soc., 68 (1978)), 140-142 · Zbl 0376.20016 |
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