4-Engel implies locally nilpotent for groups

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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 (i.e., 4-Engel group) must also satisfy the second group property (i.e., locally nilpotent group)
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The statement has the following equivalent formulations:

  1. Any 4-Engel group is a locally nilpotent group.
  2. Any finitely generated 4-Engel group is a nilpotent group. (In fact, we can work out an explicit bound on the nilpotency class of the group in terms of the size of the generating set).

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