Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-15T12:12:55.909Z Has data issue: false hasContentIssue false

On 𝓕-subnormal subgroups and Frattini-like subgroups of a finite group

Published online by Cambridge University Press:  18 May 2009

A. Ballester-Bolinches
Affiliation:
Departament D'Algebra, Universitat de Valencia, C/Dr. Moliner 50, 46100 Burjassot (Valencia), Spain
M. D. PĂ©rez-Ramos
Affiliation:
Departament D'Algebra, Universitat de Valencia, C/Dr. Moliner 50, 46100 Burjassot (Valencia), Spain
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Throughout the paper we consider only finite groups.

J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Ω(G), the Frattini subgroup of G, such that H/Ω(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Ballester-Bolinches, A., Maximal subgroups and formations. J. Pure Appl. Algebra 61 (1989), 223–232.Google Scholar
2.Ballester-Bolinches, A., Doerk, K. and PĂ©rez-Ramos, M. D., On 𝓕normal subgroups of finite soluble groups, preprint.Google Scholar
3.Beidleman, J. C. and Smith, H., On Frattini-like subgroups, Glasgow Math. J. 35 (1993), 95–98.Google Scholar
4.Bhattacharya, P. and Mukherjee, N. P., On the intersection of a class of maximal subgroups of a finite group II, J. Pure Appl. Algebra 42 (1986), 117–124.Google Scholar
5.Doerk, K. and Hawkes, T. O., Finite soluble groups (De Gruyter, Berlin-New York, 1992).Google Scholar
6.Feng, Y. and Zhang, B., Frattini subgroups relative to formation functions, J. Pure Appl. Algebra 64 (1990), 145–148.Google Scholar
7.Förster, P., On finite groups all of whose subgroups are 𝓕-subnormal or 𝓕subabnormal, J. Algebra 103 (1986), 285–293.Google Scholar
8.Griess, R. L. and Schmid, P., The Frattini module, Arch. Math. 30 (1978), 256–266.Google Scholar
9.Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, 1967).Google Scholar