# Kernel of a bihomomorphism

From Groupprops

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

for some group , such that:

## 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 |