Geodesic
The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group
Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 41-46

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

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. 
Beidleman, James C. The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group. Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 41-46. doi: 10.4153/CJM-1970-005-1
@article{10_4153_CJM_1970_005_1,
     author = {Beidleman, James C.},
     title = {The {Influence} of {Generalized} {Frattini} {Subgroups} on the {Solvability} of a {Finite} {Group}},
     journal = {Canadian journal of mathematics},
     pages = {41--46},
     year = {1970},
     volume = {22},
     number = {1},
     doi = {10.4153/CJM-1970-005-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-005-1/}
}
TY  - JOUR
AU  - Beidleman, James C.
TI  - The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group
JO  - Canadian journal of mathematics
PY  - 1970
SP  - 41
EP  - 46
VL  - 22
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-005-1/
DO  - 10.4153/CJM-1970-005-1
ID  - 10_4153_CJM_1970_005_1
ER  - 
%0 Journal Article
%A Beidleman, James C.
%T The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group
%J Canadian journal of mathematics
%D 1970
%P 41-46
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-005-1/
%R 10.4153/CJM-1970-005-1
%F 10_4153_CJM_1970_005_1

[1] 1. Baer, R., Nilpotent characteristic subgroups of finite groups, Amer. J. Math. 75 (1953), 633–664. Google Scholar

[2] 2. Baer, R., Classes of finite groups and their properties, Illinois J. Math. 1 (1957), 115–187. Google Scholar

[3] 3. Beidleman, J. C., Generalized Frattini subgroups of finite groups. II, Can. J. Math. 21 (1969), 418–429. Google Scholar

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

[5] 5. Huppert, B., Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409–434. Google Scholar

[6] 6. Rose, J. S., The influence on a finite group of its proper abnormal structure, J. London Math. Soc. 40 (1965), 348–361. Google Scholar

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

Cité par Sources :