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Generalized Frattini Subgroups of Finite Groups. II

Published online by Cambridge University Press:  20 November 2018

James C. Beidleman*
Affiliation:
University of Kentucky, Lexington, Kentucky
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The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.

Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.

Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Baer, R., Nilpotent characteristic subgroups of finite groups, Amer. J. Math. 75 (1953), 633664.Google Scholar
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3. Bechtell, H., Pseudo-Frattini subgroups, Pacific J. Math. 44 (1964), 11291136.Google Scholar
4. Beidleman, J. and Seo, T., Generalized Frattini subgroups of finite groups, Pacific J. Math. 23 (1967), 441450.Google Scholar
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9. Taunt, D. R., On A-groups, Proc. Cambridge Philos. Soc. 45 (1949), 2442.Google Scholar
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