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.

中文翻译：

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具有完美订单分类的Frobenius组

本文的目的是研究具有“完美阶数类”的有限Frobenius群。也就是说，每个订单的元素数量是该组订单的因数的因子。如果有限的Frobenius群具有完美的序类，那么它的Frobenius补体也是如此，则Frobenius核是奇次幂阶的同环群，并且Frobenius补子会定期作用于Frobenius核的素数阶元素。反之亦然。结合基本的数论论证，我们使用它来描述Frobenius群的几个重要类别的特征。具有完美阶次类别的不溶Frobenius群得到充分表征。这些是完美的Frobenius群，其Frobenius内核是同级$ 11 $-组，排名$ 2 $。我们还精确地确定哪些幂零Frobenius补码具有完善的订单类，由此得出，只有当Frobenius补数是循环的偶数阶$ \ {2,3 \} $-组时，具有幂零补码的Frobenius组才具有完美的订单类。 。还完整描述了Frobenius补码为双主群的Frobenius群组，并且我们显示，没有Frobenius补码为$ \ {2,3,5 \} $-基团且可被$ 30 $整除的可溶Frobenius基团。完美的订单分类。