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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
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
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 41 - 46
Copyright
Copyright © Canadian Mathematical Society 1970

References

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