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.

References

• Ol'shanskiĭ, A. Yu. (1995), "SQ-universality of hyperbolic groups", Matematicheskii Sbornik, 186 (8): 119–132, Bibcode:1995SbMat.186.1199O, doi:10.1070/SM1995v186n08ABEH000063, MR 1357360.
• Sonkin, Dmitriy (2003), "CEP-subgroups of free Burnside groups of large odd exponents", Communications in Algebra, 31 (10): 4687–4695, doi:10.1081/AGB-120023127, MR 1998023, S2CID 121678772.