具有完美阶类的 Frobenius 群
Frobenius Groups with Perfect Order Classes
AI总结：
本文刻画了具有完美阶类的有限 Frobenius 群，证明其 Frobenius 核必为奇素数幂阶同循环群且补正则作用于核的素数阶元素，并完全分类了不可解情形及幂零补、双准素补等情形。
摘要：
本文旨在研究具有“完美阶类”的有限 Frobenius 群；即对于每个阶，其元素个数都是该群阶数的因子。如果一个有限 Frobenius 群具有完美阶类，那么它的 Frobenius 补也具有完美阶类，Frobenius 核是一个奇素数幂阶的同循环群，并且 Frobenius 补正则地作用于 Frobenius 核中素数阶元素上。反过来也成立。结合初等数论论证，我们利用这一结果刻画了几类重要的 Frobenius 群。具有完美阶类的不可解 Frobenius 群被完全刻画。这些群恰好是其 Frobenius 核为秩 2 的同循环 11-群的完美 Frobenius 群。我们还精确确定了哪些幂零 Frobenius 补具有完美阶类，由此得出：具有幂零补的 Frobenius 群具有完美阶类，当且仅当其 Frobenius 补是偶阶循环 {2,3}-群。Frobenius 补为双准素群的那些 Frobenius 群也被完整描述，并且我们证明：若可解 Frobenius 群的 Frobenius 补是一个阶可被 30 整除的 {2,3,5}-群，则该群不具有完美阶类。
英文摘要：
The purpose of this paper is to investigate the finite Frobenius groups with "perfect order classes"; that is, those for which the number of elements of each order is a divisor of the order of the group. If a finite Frobenius group has perfect order classes then so too does its Frobenius complement, the Frobenius kernel is a homocyclic group of odd prime power order, and the Frobenius complement acts regularly on the elements of prime order in the Frobenius kernel. The converse is also true. Combined with elementary number-theoretic arguments, we use this to provide characterisations of several important classes of Frobenius groups. The insoluble Frobenius groups with perfect order classes are fully characterised. These turn out to be the perfect Frobenius groups whose Frobenius kernel is a homocyclic $11$-group of rank $2$. We also determine precisely which nilpotent Frobenius complements have perfect order classes, from which it follows that a Frobenius group with nilpotent complement has perfect order classes only if the Frobenius complement is a cyclic $\{2,3\}$-group of even order. Those Frobenius groups for which the Frobenius complement is a biprimary group are also described fully, and we show that no soluble Frobenius group whose Frobenius complement is a $\{2,3,5\}$-group with order divisible by $30$ has perfect order classes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
