FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 2, PAGES 121-131

Finite solvable groups in which the Sylow $p$-subgroups are either bicyclic or of order $p3$

V. S. Monakhov
A. A. Trofimuk

Abstract

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All groups considered in this paper will be finite. Our main result here is the following theorem. Let $G$ be a solvable group in which the Sylow $p$-subgroups are either bicyclic or of order $p3$ for any $p$Î p (G). Then the derived length of $G$ is at most $6$. In particular, if $G$ is an $A$₄-free group, then the following statements are true: (1) $G$ is a dispersive group; (2) if no prime $q$Î p (G) divides $p2+p+1$ for any prime $p$Î p (G), then $G$ is Ore dispersive; (3) the derived length of $G$ is at most $4$.

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Last modified: December 22, 2009