Normal embeddings of p-groups into p-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.