A nonempty class $$\mathfrak {X}$$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We consider two definitions of submaximal $$\mathfrak {X}$$ -subgroups suggested by H. Wielandt and discuss which one better suits the task of determining maximal $$\mathfrak {X}$$ -subgroups. We prove that these definitions are not equivalent yet the Wielandt–Hartley theorem holds true for either definition of $$\mathfrak {X}$$ -submaximality. We also give some applications of the strong version of the Wielandt–Hartley theorem.

中文翻译：

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次极大 $$\mathfrak {X}$$-子群的 Wielandt–Hartley 定理

如果有限群的非空类 $$\mathfrak {X}$$ 在取子群、同态图像和扩展下是封闭的，则称其为完全类。我们考虑 H. Wielandt 建议的次极大 $$\mathfrak {X}$$ -subgroups 的两种定义，并讨论哪一个更适合确定最大 $$\mathfrak {X}$$ -subgroups 的任务。我们证明这些定义并不等价，但 Wielandt-Hartley 定理对于 $$\mathfrak {X}$$ -submaximality 的任一定义都成立。我们还给出了 Wielandt-Hartley 定理的强版本的一些应用。