# Homomorph-containment is quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., homomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)

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

Get more facts about homomorph-containing subgroup |Get facts that use property satisfaction of homomorph-containing subgroup | Get facts that use property satisfaction of homomorph-containing subgroup|Get more facts about quotient-transitive subgroup property

## Contents

## Statement

### Statement with symbols

Suppose is a group, and are subgroups of such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of . Then, is a homomorph-containing subgroup of .

## Related facts

### Related facts about homomorph-containing subgroups

- Homomorph-containment is not transitive
- Homomorph-containment satisfies intermediate subgroup condition