Free implies residually nilpotent
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
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Further information: free group
Residually nilpotent group
Further information: residually nilpotent group
Pick a freely generating set for the free group. Then, we show that the length of the shortest word for any non-identity 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 .