# Pure definability is quotient-transitive

From Groupprops

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

## Statement

### Statement with symbols

Suppose are groups such that is a purely definable subgroup of and is a purely definable subgroup of (note that any purely definable subgroup is characteristic and hence normal, so it makes sense to take the quotient group). Then, is a purely definable subgroup of .