Volume 34 • Number 2 • 2011
 • Finite Groups in which Primary Subgroups have Cyclic Cofactors Yufeng Liu and Xiaolan Yi Abstract. In this paper, we prove the following theorem: Let $G$ be a group, $q$ be the largest prime divisor of $|G|$ and $\pi =\pi (G)\setminus \{q\}$. Suppose that the factor group $X/core_GX$ is cyclic for every $p$-subgroup $X$ of $G$ and every $p\in \pi$. Then:     (1). $G$ is soluble and its Hall $\{2, 3\}'$-subgroup is normal in $G$ and is a dispersive group by Ore;     (2). All Hall $\{2, 3\}$-subgroups of $G$ are metanilpotent;     (3). Every Hall $p'$-subgroup of $G$ is a dispersive group by Ore, for every $p\in \{2, 3\}$;     (4). $l_{r}(G)\leq1$, for all $r\in \pi (G)$. 2010 Mathematics Subject Classification: 20D10, 20D20. Full text: PDF