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