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 Finite $p$-groups all of whose subgroups of class 2 are generated by two elements J. Korean Math. Soc.Published online 2019 Feb 01 Pujin Li, and Qinhai Zhang Shanxi Normal University Abstract : We proved that finite $p$-groups in the title coincide with finite $p$-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite $p$-group all of whose class 2 are minimal non-abelian(of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved. Keywords : finite $p$-groups, minimal non-abelian $p$-groups, subgroups of class 2 MSC numbers : 20D15; 20F05 Full-Text :