Finite group admitting a bijective quasihomomorphism to an abelian group

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

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