On elementary groups
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- by Ernest L. Stitzinger PDF
- Proc. Amer. Math. Soc. 26 (1970), 236-238 Request permission
Erratum: Proc. Amer. Math. Soc. 34 (1972), 631.
Abstract:
Bechtell has defined a group $G$ to be elementary if the Frattini subgroup of each subgroup of $G$ is the identity. In this note we prove the following: If the derived group of $G$ is nilpotent, then necessary and sufficient conditions that $G$ be elementary are that the Frattini subgroup of $G$ be the identity and that the Frattini subgroup of some Carter subgroup $K$ of $G$ be equal to the derived group of $K$.References
- Homer Bechtell, Elementary groups, Trans. Amer. Math. Soc. 114 (1965), 355–362. MR 175967, DOI 10.1090/S0002-9947-1965-0175967-3
- Roger W. Carter, Nilpotent self-normalizing subgroups of soluble groups, Math. Z. 75 (1960/61), 136–139. MR 123603, DOI 10.1007/BF01211016
- Wolfgang Gaschütz, Über die $\Phi$-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170 (German). MR 57873, DOI 10.1007/BF01174137
- W. R. Scott, Group theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0167513
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 26 (1970), 236-238
- MSC: Primary 20.40
- DOI: https://doi.org/10.1090/S0002-9939-1970-0265467-1
- MathSciNet review: 0265467