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The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group

Published online by Cambridge University Press:  20 November 2018

James C. Beidleman*
Affiliation:
University of Kentucky, Lexington, Kentucky
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1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Baer, R., Nilpotent characteristic subgroups of finite groups, Amer. J. Math. 75 (1953), 633664.Google Scholar
2. Baer, R., Classes of finite groups and their properties, Illinois J. Math. 1 (1957), 115187.Google Scholar
3. Beidleman, J. C., Generalized Frattini subgroups of finite groups. II, Can. J. Math. 21 (1969), 418429.Google Scholar
4. Beidleman, J. C. and Seo, T. K., Generalized Frattini subgroups of finite groups, Pacific J. Math. 23 (1967), 441450.Google Scholar
5. Huppert, B., Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409434.Google Scholar
6. Rose, J. S., The influence on a finite group of its proper abnormal structure, J. London Math. Soc. 40 (1965), 348361.Google Scholar
7. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar
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