Normal subgroups contained in Frattini subgroups are Frattini subgroups

Allenby, R. B. J. T.

doi:10.1090/S0002-9939-1980-0553365-1

Normal subgroups contained in Frattini subgroups are Frattini subgroups
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by R. B. J. T. Allenby PDF

Proc. Amer. Math. Soc. 78 (1980), 315-318 Request permission

Abstract:

We prove that if N is a normal subgroup of the finite group G and if $N \subseteq \Phi (G)$, then there exists a finite group U such that $N = \Phi (U)$ exactly. In particular, we see that the generalizations apparent in the conclusions of several recently stated theorems are illusory.

References

R. B. J. T. Allenby and C. Y. Tang, On the Frattini subgroups of generalized free products, J. Algebra 52 (1978), no. 2, 460–474. MR 480759, DOI 10.1016/0021-8693(78)90250-8
Wolfgang Gaschütz, Über die $\Phi$-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170 (German). MR 57873, DOI 10.1007/BF01174137
W. Mack Hill and Donald B. Parker, The nilpotence class of the Frattini subgroup, Israel J. Math. 15 (1973), 211–215. MR 322058, DOI 10.1007/BF02787567
W. Mack Hill and Charles R. B. Wright, Normal subgroups contained in the Frattini subgroup, Proc. Amer. Math. Soc. 35 (1972), 413–415. MR 301094, DOI 10.1090/S0002-9939-1972-0301094-7
Alexander Kurosch, Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934), no. 1, 647–660 (German). MR 1512914, DOI 10.1007/BF01449159
B. H. Neumann, Ascending verbal and Frattini series, Math. Z. 69 (1958), 164–172. MR 96735, DOI 10.1007/BF01187397
W. R. Scott, Group theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0167513
Wilhelm Specht, Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956 (German). MR 0080091
Ernest L. Stitzinger, A nonembedding theorem for finite groups, Proc. Amer. Math. Soc. 25 (1970), 124–126. MR 258936, DOI 10.1090/S0002-9939-1970-0258936-1
Helmut Wielandt, Einbettung zweier Gruppen in eine einfache Gruppe, Math. Z. 73 (1960), 20–21 (German). MR 124387, DOI 10.1007/BF01163266