关于有限<italic>p</italic>-群的正规闭包的一些条件

Abstract

Berkovich posed the following problem: study the finite <italic>p</italic>-group <italic>G</italic> which satis es the conditions that, for each non-normal subgroup<italic> H</italic> of <italic>G</italic>, (<italic>N</italic>₁) exp(<italic>H</italic>)=exp(<italic>H^G</italic>); (<italic>N</italic>₂) |<italic>H^G</italic>:<italic>H|</italic>≤<italic>p</italic>; (<italic>N</italic>₃) <italic>H^G</italic>=<italic>HG</italic>. In this paper, we first study the finite <italic>p</italic>-groups satisfying Condition (<italic>N</italic>₃), and then study the finite p-groups satisfying Conditions (<italic>N</italic>₁); (<italic>N</italic>₂) and (<italic>N</italic>₃).