The Conciseness Of Words In The Residually Finite Groups

We guess the words are boundedly concise in the residually finite groups add to prime power q is still boundedly concise.A word imply virtual nilpotency is boundedly concise in the residually finite groups.In this article,we proved that if word ω imply virtual nilpotency and q is a prime power,ωq is boundedly concise in the residually finite groups.It remains unkown whether for all concise words ω,ωq is boundely concise in the residually finite groups.This paper is divided into four chapters.The first chapter,introduction,introduces some concepts about the conciseness of words and the unresolved problems about words,and through the literature known concise words in a variety class of groups and draw our own ideas.In the second chapter,some concise words in the residually finite groups are sorted out,and some definitions,theorems and lemmas about concise and boundely concise words are introduced.In chapter 3,on the basis of chapter 2,the concise words in the residually finite groups and some theorems to prove conciseness are studied in detail.Based on these revelations,new concise words in the residually finite groups are constructed.By thinking about the theorems,I found that I mainly focused on the concise words about the prime power q,which could be linked to p-group,and I could consider the proof method of the theorem in many aspects.In chapter 4,in order to make the proof of the main theorem short,in the previous revelation,we first introduce a lemma,apply the properties of several groups and always consider the quotient group,and find a p-group,the lemma is proved.The main theorem is proved under some basis group knowledge and lemmas.