Free implies residually nilpotent
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
View all group property implications  View all group property nonimplications

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Contents
Statement
Verbal statement
Any free group is residually nilpotent group: the intersection of the terms of its finite lower central series is trivial.
Definitions used
Free group
Further information: free group
Residually nilpotent group
Further information: residually nilpotent group
Proof
Proof outline
Pick a freely generating set for the free group. Then, we show that the length of the shortest word for any nonidentity element in the term of the lower central series, is bounded from below by . Thus, if an element of the free group can be expressed as a word of length , it cannot occur in the member for .