Let G be a finite group of odd order admitting an involutory automorphism $$\phi $$ , and let $$G_{-\phi }$$ be the set of elements of G transformed by $$\phi $$ into their inverses. Note that $$[G,\phi ]$$ is precisely the subgroup generated by $$G_{-\phi }$$ . Suppose that each subgroup generated by a subset of $$G_{-\phi }$$ can be generated by at most r elements. We show that the rank of $$[G,\phi ]$$ is r-bounded.

中文翻译：

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关于具有对合自同构的奇数阶有限群的秩

设 G 是一个允许对合自同构 $$\phi $$ 的奇阶有限群，并令 $$G_{-\phi }$$ 是 G 的由 $$\phi $$ 变换成它们的逆的元素的集合. 请注意， $$[G,\phi ]$$ 正是 $$G_{-\phi }$$ 生成的子群。假设由$$G_{-\phi }$$ 的子集生成的每个子群最多可以由r 个元素生成。我们证明 $$[G,\phi ]$$ 的秩是 r 有界的。