Finite group admitting a bijective quasihomomorphism to an abelian group

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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Definition

A finite group $G$ is termed a finite group admitting a bijective quasihomomorphism to an abelian group if there is an abelian group (in particular, a finite abelian group) $H$ and a bijective function $f:G \to H$ that is a quasihomomorphism of groups: whenever $g_1, g_2 \in G$ commute, we have: $f(g_1g_2) = f(g_1)f(g_2)$

Equivalently, it is a finite nilpotent group and each of its Sylow subgroups is a group of prime power order admitting a bijective quasihomomorphism to an abelian group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite Lazard Lie group
Finite abelian group
Group of prime power order admitting a bijective quasihomomorphism to an abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group that is 1-isomorphic to an abelian group
Finite group that is order statistics-equivalent to an abelian group