Some corollaries of Frobeniusβ normal $p$-complement theorem
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- by Yakov Berkovich PDF
- Proc. Amer. Math. Soc. 127 (1999), 2505-2509 Request permission
Abstract:
For a prime divisor $q$ of the order of a finite group $G$, we present the set of $q$-subgroups generating $\text {O}^{q,qβ}(G)$. In particular, we present the set of primary subgroups of $G$ generating the last member of the lower central series of $G$. The proof is based on the Frobenius Normal $p$-Complement Theorem and basic properties of minimal nonnilpotent groups. Let $G$ be a group and $\Theta$ a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non-$\Theta$-subgroups ($=\Theta _{1}$-subgroups) of $G$ are not nilpotent. Then (see the lemma), if $K$ is generated by all $\Theta _{1}$-subgroups of $G$ it follows that $G/K$ is a $\Theta$-group.References
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Additional Information
- Yakov Berkovich
- Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
- Email: berkov@mathcs2.haifa.ac.il
- Received by editor(s): May 14, 1997
- Published electronically: April 28, 1999
- Additional Notes: The author was supported in part by the Ministry of Absorption of Israel.
- Communicated by: Ronald M. Solomon
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2505-2509
- MSC (1991): Primary 20D20
- DOI: https://doi.org/10.1090/S0002-9939-99-05275-2
- MathSciNet review: 1657758