# Strict characteristicity is quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., strictly characteristic 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 strictly characteristic subgroup |Get facts that use property satisfaction of strictly characteristic subgroup | Get facts that use property satisfaction of strictly characteristic subgroup|Get more facts about quotient-transitive subgroup property

## Statement

Suppose are groups such that is a strictly characteristic subgroup of and is a strictly characteristic subgroup of the quotient group . Then, is a strictly characteristic subgroup of .

## Related facts

### Generalizations

Quotient-balanced implies quotient-transitive. Other special cases of this are:

- Characteristicity is quotient-transitive
- Normality is quotient-transitive
- Full invariance is quotient-transitive

## Proof

**Given**: Groups such that is strictly characteristic in and is strictly characteristic in .

**To prove**: is strictly characteristic in .