Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

中文翻译：

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卡特子群上的 Frobenius 作用

令 [Formula: see text] 为有限可解群， [Formula: see text] 为 [Formula: see text] 的子群。假设存在 [Formula: see text] 的 [Formula: see text] 不变的 Carter 子群 [Formula: see text] 使得半直积 [Formula: see text] 是具有核 [Formula: see text] 的 Frobenius 群]和补充[公式：见正文]。我们证明[公式：见文]的拟合级数项是[公式：见文]与[公式：见文]的拟合级数对应项的交集，得到[公式：见文]的拟合高度。公式：见文]最多可以超过[公式：见文]的拟合高度一。作为推论，它表明对于任何素数集[公式：参见文本]，[公式：参见文本]系列的项[公式：see text] 是 [Formula: see text] 与 [Formula: see text]-series of [Formula: see text] 的对应项与 [Formula: see text]-length 的交集: see text] 最多可以超过 [Formula: see text] 的 [Formula: see text] 长度 1。这些定理概括了 Khukhro 获得的 [EI Khukhro, Fitting height of afinite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1-11] 中的主要结果。