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
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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)	|FULL LIST, MORE INFO
group of nilpotency class two	nilpotency class at most two	class two implies commutator map is endomorphism		|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