Geodesic
Generalized Frattini Subgroups of Finite Groups. II
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 418-429

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Beidleman, James C. Generalized Frattini Subgroups of Finite Groups. II. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 418-429. doi: 10.4153/CJM-1969-046-3
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[1] 1. Baer, R., Nilpotent characteristic subgroups of finite groups, Amer. J. Math. 75 (1953), 633–664. Google Scholar

[2] 2. Bechtell, H., Elementary groups, Trans. Amer. Math. Soc. 114 (1965), 355–362. Google Scholar

[3] 3. Bechtell, H., Pseudo-Frattini subgroups, Pacific J. Math. 44 (1964), 1129–1136. Google Scholar

[4] 4. Beidleman, J. and Seo, T., Generalized Frattini subgroups of finite groups, Pacific J. Math. 23 (1967), 441–450. Google Scholar

[5] 5. Deskins, W. E., A condition for the solvability of a finite group, Illinois J. Math. 5 (1961), 306–313. Google Scholar

[6] 6. Hall, P., Complemented groups, J. London Math. Soc. 12 (1937), 201–204. Google Scholar

[7] 7. Hall, P., On system normalizers of a solvable group, Proc. London Math. Soc. 43 (1937), 507–528. Google Scholar

[8] 8. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1964). Google Scholar

[9] 9. Taunt, D. R., On A-groups, Proc. Cambridge Philos. Soc. 45 (1949), 24–42. Google Scholar

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