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<p>We study the following conjecture, which is a sharp analogue of the well-known Baer–Suzuki theorem for the π-radical of a finite group. For an arbitrary set π of primes not containing all primes, let r be the smallest prime not in π. Set m = r if r ⩽ 3 and m = r − 1 if r &gt; 3. Then, in a finite group <i>G</i>, the largest normal π-subgroup always coincides with the set of elements <i>x</i> such that any m conjugates of <i>x</i> generate a π-subgroup. To date, this conjecture has been confirmed for any finite group whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups <sup>2</sup><i>B</i><sub>2</sub>(<i>q</i>), <sup>2</sup><i>G</i><sub>2</sub>(<i>q</i>), <sup>2</sup><i>F</i><sub>4</sub>(<i>q</i>)′, <i>G</i><sub>2</sub>(<i>q</i>), or <sup>3</sup><i>D</i><sub>4</sub>(<i>q</i>). It is proved that the simple symplectic groups <i>S</i><sub>2n</sub>(<i>q</i>) can be added to this list.</p>

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Towards the Sharp Baer–suzuki Theorem for the π-radical: Symplectic Groups

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • D. O. Revin

摘要

We study the following conjecture, which is a sharp analogue of the well-known Baer–Suzuki theorem for the π-radical of a finite group. For an arbitrary set π of primes not containing all primes, let r be the smallest prime not in π. Set m = r if r ⩽ 3 and m = r − 1 if r > 3. Then, in a finite group G, the largest normal π-subgroup always coincides with the set of elements x such that any m conjugates of x generate a π-subgroup. To date, this conjecture has been confirmed for any finite group whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups 2B2(q), 2G2(q), 2F4(q)′, G2(q), or 3D4(q). It is proved that the simple symplectic groups S2n(q) can be added to this list.