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 ${\displaystyle H}$ is a CEP subgroup in a group ${\displaystyle G}$ if every normal subgroup ${\displaystyle N}$ of ${\displaystyle H}$ can be realized as ${\displaystyle H\cap M}$ where ${\displaystyle M}$ is normal in ${\displaystyle G}$.

The following facts are known about CEP subgroups:

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