Normal embeddings of <i>p</i>-groups into <i>p</i>-groups

Abstract

A well known lemma of Burnside is generalised, to give necessary and sufficient conditions for a finite p -group K to be normally embedded in a nilpotent group V , with K ⊆ω( V ). (Here, ω denotes a single word and ω( V ) is the corresponding verbal subgroup.) Our main result is related to earlier work of Blackburn, Gaschütz and Hobby.