CEP-subgroup

Symbol-free definition
A subgroup of a group is termed a CEP subgroup (or a group with Congruence Extension Property) if any normal subgroup of this subgroup is the intersection of this subgroup with a normal subgroup of the whole group.

CEP-subgroups are also termed normal-sensitive subgroups.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a CEP subgroup if for any normal subgroup $$N$$ of $$H$$, there is a normal subgroup $$M$$ of $$G$$ such that $$N = M$$ &cap; $$H$$.

In terms of the subgroup intersection extension formalism
In terms of the subgroup intersection extension formalism, the property of being a CEP subgroup is the balanced property with respect to the subgroup property of normality.

Facts
Every subgroup of a group is a CEP-subgroup if and only if the group is a solvable T-group.

Stronger properties

 * Weaker than::Retract
 * Weaker than::Transitively normal subgroup

Weaker properties

 * Stronger than::Subgroup in which every relatively normal subgroup is strongly closed
 * Stronger than::Subgroup in which every relatively normal subgroup is weakly closed

Metaproperties
Any CEP subgroup of a CEP subgroup is a CEP subgroup. This follows from the fact that the property of being CEP is a balanced subgroup property with respect to a suitable formalism.

The improper subgroup, or the whole group, is clearly a CEP subgroup, so the property of being a CEP subgroup is identity-true.

The trivial subgroup is also a CEP subgroup, so the property of being a CEP subgroup is trivially true.

Thus, the property of being a CEP subgroup is a trim subgroup property.