CEP subgroup

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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 normal in a group G if every normal subgroup N of H can be realized as 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.