Journal of Group Theory ( IF 0.466 ) Pub Date : 2020-09-09 , DOI: 10.1515/jgth-2019-0159
Lihua Zhang; Junqiang Zhang

Assume 𝐺 is a finite 𝑝-group. We prove that if all $A2$-subgroups of 𝐺 are generated by two elements, then so are all non-abelian subgroups of 𝐺. By using this result, we classify the 𝑝-groups which have at least two $A1$-subgroups and in which the intersection of every pair of distinct $A1$-subgroups equals the intersection of all the $A1$-subgroups. It turns out that such 𝑝-groups are the finite 𝑝-groups all of whose non-abelian subgroups are generated by two elements, and the converse also almost holds.

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