# Group in which order of commutator divides order of element

## Definition

A group in which order of commutator divides order of element is a group where, for any elements $x$ and $y$ such that $x$ has finite order, the commutator $[x,y]$ also has finite order and the order of $[x,y]$ divides the order of $x$.

Note that the definition does not depend on whether we use the left definition of commutator ( $[x,y] = xyx^{-1}y^{-1}$) or the right definition ( $[x,y] = x^{-1}y^{-1}xy$).

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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Examples

### Examples among finite 2-groups

Any group of nilpotency class two satisfies this property, because class two implies commutator map is endomorphism. We list here the finite p-groups of small order that have class greater than two but satisfy the property:

Order List of groups List of GAP IDs (second part) Nilpotency class
16 generalized quaternion group:Q16 9 3
32 faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8), SmallGroup(32,10), semidirect product of Z8 and Z4 of semidihedral type, semidirect product of Z8 and Z4 of dihedral type, direct product of Q16 and Z2 6,7,8,10,13,14,15,41 3 (all of them)
64 4, 5, 7, 9, 11, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 30, 32, 33, 35, 37, 45, 49, 90, 91, 92, 93, 94, 96, 100, 106, 107, 108, 109, 110, 111, 120, 122, 132, 138, 139, 143, 148, 151, 156, 158, 160, 164, 165, 166, 168, 172, 175, 179, 180, 181, 182, 252 IDs 32, 33, 35, 37, 49 all have class four; all other IDs have class three

### Examples among finite p-groups for other primes p

Any Lazard Lie group automatically satisfies this property. In particular, any p-group of nilpotency class less than p automatically satisfies this property.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group any two elements commute (obvious) (via class two) 2-Engel group|FULL LIST, MORE INFO
group of nilpotency class two nilpotency class at most two class two implies commutator map is endomorphism 2-Engel group|FULL LIST, MORE INFO
Levi group any two elements generate a subgroup of class at most two |FULL LIST, MORE INFO
group of prime exponent all non-identity elements have order $p$ for a single prime $p$ |FULL LIST, MORE INFO
aperiodic group no non-identity element has finite order |FULL LIST, MORE INFO
Lazard Lie group group that admits a Lazard Lie ring via the Lazard correspondence |FULL LIST, MORE INFO
p-group of nilpotency class less than p nilpotent p-group whose nilpotency class is at most $p-1$ |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which elements of coprime finite orders commute any two elements of finite orders that are relatively prime to each other must commute order of commutator divides order of element implies elements of coprime finite orders commute |FULL LIST, MORE INFO
group in which every finite subgroup is nilpotent any finite subgroup of the group is nilpotent |FULL LIST, MORE INFO