Kernel of a bihomomorphism

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Suppose G is a group and H is a subgroup of G. We say that H is a kernel of a bihomomorphism in G if there exists a bihomomorphism:

b:G \times G \to M

for some group M, such that:

H = \{ x \in G \mid b(x,y) \mbox{ is the identity element of } M \mbox{ for all } y \in G\}

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian-quotient subgroup the quotient group is an abelian group. kernel of a bihomomorphism implies abelian-quotient abelian-quotient not implies kernel of a bihomomorphism Intersection of kernels of bihomomorphisms|FULL LIST, MORE INFO
completely divisibility-closed subgroup kernel of a bihomomorphism is completely divisibility-closed Intersection of kernels of bihomomorphisms|FULL LIST, MORE INFO
intersection of kernels of bihomomorphisms intersection of subgroups, each of which is a kernel of a bihomomorphism. |FULL LIST, MORE INFO