Exponent three implies class three for groups: Difference between revisions
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Suppose <math>G</math> is a [[group]] whose [[exponent of a group|exponent]] is three. Then, <math>G</math> is a [[group of nilpotency class three]]: it is a [[nilpotent group]] and its [[nilpotency class]] is at most three. | Suppose <math>G</math> is a [[group]] whose [[exponent of a group|exponent]] is three. Then, <math>G</math> is a [[group of nilpotency class three]]: it is a [[nilpotent group]] and its [[nilpotency class]] is at most three. | ||
==Related facts== | |||
* [[Exponent two implies abelian]], which follows from [[square map is endomorphism iff abelian]] | |||
==Facts used== | ==Facts used== | ||
Latest revision as of 21:57, 4 June 2012
Statement
Suppose is a group whose exponent is three. Then, is a group of nilpotency class three: it is a nilpotent group and its nilpotency class is at most three.
Related facts
- Exponent two implies abelian, which follows from square map is endomorphism iff abelian
Facts used
- Exponent three implies 2-Engel for groups (note that the analogous statement is not true for Lie rings)
- 2-Engel implies class three for groups (note that the analogous statement is true for Lie rings)
Proof
The proof follows directly by combining Facts (1) and (2).