# Homomorph-containment is strongly join-closed

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., strongly join-closed subgroup property)

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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 strongly join-closed subgroup property

## Contents

## Statement

### Statement with symbols

Suppose is a group, is an indexing set, and , is a collection of homomorph-containing subgroups of . Then, the join of subgroups is also a homomorph-containing subgroup of .

## Related facts

### Related facts about join-closed

- Isomorph-containment is strongly join-closed
- Isomorph-freeness is strongly join-closed
- Full invariance is strongly join-closed
- Characteristicity is strongly join-closed
- Normality is strongly join-closed

## Proof

**Given**: A group , an indexing set , a collection of homomorph-containing subgroups of , . . A homomorphism .

**To prove**: is containined in .

**Proof**: . Since each is homomorph-containing in , is contained in , so the join of , is contained in the join of the s, which is .