CEP subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a congruence of the whole group. Equivalently, every normal subgroup of the subgroup arises as the intersection with the subgroup of a normal subgroup of the whole group.

In symbols, a subgroup $$H$$ is a CEP subgroup in a group $$G$$ if every normal subgroup $$N$$ of $$H$$ can be realized as $$H \cap M$$ where $$M$$ is normal in $$G$$.

The following facts are known about CEP subgroups:


 * Every retract has the CEP.
 * Every transitively normal subgroup has the CEP.