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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 ${\displaystyle H}$ 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.

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.

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