# 2-Engel group

From Groupprops

## Definition

No. | Shorthand | A group is termed a Levi group or a 2-Engel group if ... | A group is termed a Levi group or 2-Engel group if ... |
---|---|---|---|

1 | conjugates commute | any two conjugate elements of the group commute. | commutes with for all |

2 | normal closures abelian | the normal closure of any cyclic subgroup (or the normal subgroup generated by any one-element subset) is abelian | the normal subgroup generated by is abelian for all . |

3 | union of abelian normal subgroups | the group is a union (as a set) of abelian normal subgroups |
there is a collection of abelian normal subgroups of such that |

4 | 2-local class two | the 2-local nilpotency class of the group is at most 2. | for any , the subgroup is a group of class at most two. |

5 | 2-Engel | the group is a -Engel group: the commutator between any element and its commutator with another element is the identity element. | the commutator is the identity element for all . |

6 | cyclic property of triple commutators | triple commutators are preserved under cyclic permutation of the inputs. | for all , we have . |

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Formalisms

### In terms of the Levi operator

This property is obtained by applying the Levi operator to the property: Abelian group

View other properties obtained by applying the Levi operator

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group | 3-abelian group|FULL LIST, MORE INFO | |||

Dedekind group | every subgroup is normal | |FULL LIST, MORE INFO | ||

group of nilpotency class two | nilpotency class at most two; or, quotient by center is an abelian group | 2-Engel not implies class two for groups | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group generated by abelian normal subgroups | generated by abelian normal subgroups | |FULL LIST, MORE INFO | ||

bounded Engel group | -Engel group for some finite | |FULL LIST, MORE INFO | ||

Engel group | For any two elements and , the iterated commutator of with eventually becomes trivial | |FULL LIST, MORE INFO | ||

group in which order of commutator divides order of element | For any two elements and , if the order of is finite, the order of divides the order of | |FULL LIST, MORE INFO | ||

nilpotent group (for finite groups) |