title: CEP subgroup
text: # 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.
  