# 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