# Kernel of a bihomomorphism

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