4-Engel and (2,3,5)-torsion-free implies class seven for groups

Statement

Suppose $G$ is a 4-Engel group. Suppose, further, that $G$ does not have any 2-torsion, 3-torsion, or 5-torsion, i.e., $G$ does not have any non-identity element of order 2, 3, or 5. Then, $G$ is a nilpotent group and its nilpotency class is at most seven.

Related facts

Similar facts for 4-Engel groups

• 4-Engel implies locally nilpotent

Similar facts for other Engel groups

• 2-Engel implies class three for groups
• 2-Engel and 3-torsion free implies class two for groups
• 3-Engel and (2,5)-torsion-free implies class four for groups