# 3-local nilpotency class

## Definition

Suppose is a group. The 3-local nilpotency class is defined as the 3-local nilpotency class of . Explicitly, it is the supremum, over 3-generated subgroups of , of the nilpotency class of . In other words, it is defined as:

(Note that are allowed to be equal to each other, but this does not matter for nontrivial groups).

If there is a non-nilpotent subgroup of generated by three or fewerelements, then is not 3-locally nilpotent. It is also possible that be non-nilpotent because, while each 3-generated subgroup is nilpotent, there is no upper bound on the nilpotency class. An example is the generalized dihedral group for 2-quasicyclic group.

Note that when we say "a group of 3-local nilpotency class " we usually mean "a group whose 3-local nilpotency class is at *most* ."

In general, the 3-local nilpotency class of a nilpotent group is less than or equal to its nilpotency class.

## Related notions

## Particular cases

3-local nilpotency class | What it tells us about the group |
---|---|

1 | The group is an abelian group. This follows from the observation that abelianness is 2-local. |

2 | This is the same as being a group of nilpotency class two. |

3 | Unclear, but the corresponding notion for Lie rings is clear: for a Lie ring, having 3-local nilpotency class at most three is equivalent to having (global) nilpotency class at most three. See nilpotency class three is 3-local for Lie rings. |