Exponent three implies class three for groups

From Groupprops
Jump to: navigation, search

Statement

Suppose G is a group whose exponent is three. Then, G is a group of nilpotency class three: it is a nilpotent group and its nilpotency class is at most three.

Related facts

Facts used

  1. Exponent three implies 2-Engel for groups (note that the analogous statement is not true for Lie rings)
  2. 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).