3-Engel implies locally nilpotent for groups: Difference between revisions
(Created page with "{{group property implication| stronger = 3-Engel group| weaker = locally nilpotent group}} ==Statement== The statement has the following equivalent formulations: # Any [[3-...") |
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# Any [[3-Engel group]] is a [[locally nilpotent group]]. | # Any [[3-Engel group]] is a [[locally nilpotent group]]. | ||
# Any [[finitely generated group|finitely generated]] 3-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). | # Any [[finitely generated group|finitely generated]] 3-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). | ||
==Related facts== | |||
* [[3-Engel implies locally nilpotent for Lie rings]] | |||
* [[2-Engel implies class three for groups]], [[2-Engel implies class three for Lie rings]] | |||
* [[4-Engel implies locally nilpotent for groups]] | |||
Latest revision as of 02:09, 5 June 2012
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., 3-Engel group) must also satisfy the second group property (i.e., locally nilpotent group)
View all group property implications | View all group property non-implications
Get more facts about 3-Engel group|Get more facts about locally nilpotent group
Statement
The statement has the following equivalent formulations:
- Any 3-Engel group is a locally nilpotent group.
- Any finitely generated 3-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).