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.

翻译：本文旨在研究具有“完备阶类”的有限Frobenius群，即每个阶的元素个数为群阶的因子的群。若有限Frobenius群具有完备阶类，则其Frobenius补也具有该性质，Frobenius核为奇素数幂阶的同循环群，且Frobenius补在Frobenius核中的素数阶元素集合上正则作用。反之亦成立。结合初等数论论证，我们利用此结论刻画了几类重要的Frobenius群。完全刻画了具有完备阶类的不可解Frobenius群，这些群恰好是Frobenius核为秩2的同循环$11$-群的完备Frobenius群。我们还精确确定了哪些幂零Frobenius补具有完备阶类，由此可得：具有幂零补的Frobenius群仅当Frobenius补为偶数阶循环$\{2,3\}$-群时才具备完备阶类。同时完整描述了Frobenius补为双素数群的Frobenius群，并证明：若可解Frobenius群的Frobenius补为阶被30整除的$\{2,3,5\}$-群，则该群不具有完备阶类。