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Volume 34 • Number 2 • 2011
• Finite Groups in which Primary Subgroups have Cyclic Cofactors
Yufeng Liu and Xiaolan Yi

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.

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