# Ambivalence is quotient-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., ambivalent group) satisfying a group metaproperty (i.e., quotient-closed group property)

View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about ambivalent group |Get facts that use property satisfaction of ambivalent group | Get facts that use property satisfaction of ambivalent group|Get more facts about quotient-closed group property

## Statement

If is an ambivalent group, and is a normal subgroup of , the quotient group is also an ambivalent group.

## Related facts

## Proof

**Given**: An ambivalent group , a normal subgroup of with quotient map , and an element .

**To prove**: There exists an element such that .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let be such that . | The quotient map is, by definition, surjective. | direct. | ||

2 | There exists such that . | is ambivalent. | direct from definition. | ||

3 | The element works, i.e., . | We start with , apply to both sides, and use the properties of homomorphisms. |