We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by‐product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism.

中文翻译：

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关于有限群的广义拟合高度和不溶长度

我们证明了E. Khukhro和P. Shumyatsky关于有限群的拟合高度和不溶长度的两个猜想。作为我们方法的副产品，我们还证明了Flavell结果的推广，该结果本身推广了Wielandt的拉链引理，并提供了包含在唯一最大子群中的子群的特征。我们还得出了定理的许多结果，包括对由非强制自同构性反转的有限群的奇数阶元素集的某些应用。