# Characteristicity is quotient-transitive

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

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

## Contents

## Statement

### Property-theoretic statement

The subgroup property of being a characteristic subgroup satisfies the subgroup metaproperty of being quotient-transitive.

### Statement with symbols

Suppose are subgroups such that is a characteristic subgroup of , and is a characteristic subgroup of . Then, is a characteristic subgroup of .

## Related facts

### Similar facts

- Normality is quotient-transitive
- Full invariance is quotient-transitive
- Strict characteristicity is quotient-transitive

## Proof

**Given**: A group , subgroups such that is characteristic in , and is characteristic in

**To prove**: is characteristic in

**Proof**: We pick any automorphism of , and want to show that . For this, first observe that , so induces an automorphism on the quotient , by the rule . Call this automorphism .

Then, is an automorphism of . Since is characteristic in , . Thus, for any , , and hence, unwrapping the definition, . Thus, . Since the same holds for , we conclude that , completing the proof.