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
.