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Some corollaries of Frobenius' normal $p$-complement theorem

Author: Yakov Berkovich
Journal: Proc. Amer. Math. Soc. 127 (1999), 2505-2509
MSC (1991): Primary 20D20
Published electronically: April 28, 1999
MathSciNet review: 1657758
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Abstract: For a prime divisor $q$ of the order of a finite group $G$, we present the set of $q$-subgroups generating $\text{O}^{q,q'}(G)$. 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.

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Additional Information

Yakov Berkovich
Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Keywords: Special $p$-group, minimal nonnilpotent (nonabelian, noncyclic, nonsolvable) group, $p$-nilpotent group, $p$-closed group, $\text{S}(p, q)$-group, $\text{B}(p, q)$-group
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
Article copyright: © Copyright 1999 American Mathematical Society

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