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

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, doi:10.1070/SM1995v186n08ABEH000063, Bibcode: 1995SbMat.186.1199O, http://mi.mathnet.ru/eng/msb63 .
• 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 .