Let G be a group which is generated by the set of its involutions, and assume that the set of integers which occur as orders of products of two involutions in G is \(\{1,2,3,4\}\). It is shown that \(G\simeq \textrm{PSL}(2,7)\) or \(G\simeq \textrm{PSU}(3,3)\) or G is a \(\{2,3\}\)-group and \(G/O_{2}(G) \simeq S_{3}\).

中文翻译：

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所有对合均为 3 横移的组

设G是由其对合集生成的群，并假设G中作为两个对合乘积的阶数出现的整数集为\(\{1,2,3,4\}\)。证明\(G\simeq \textrm{PSL}(2,7)\)或\(G\simeq \textrm{PSU}(3,3)\)或G是\(\{2,3 \}\) -group 和\(G/O_{2}(G) \simeq S_{3}\)。