This article defines a subgroup property related to (or which arises in the context of): geometric group theory
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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A subgroup of a group is termed a CEP subgroup (or a group with Congruence Extension Property) if any normal subgroup of this subgroup is the intersection of this subgroup with a normal subgroup of the whole group.
CEP-subgroups are also termed normal-sensitive subgroups.
Definition with symbols
In terms of the subgroup intersection extension formalism
Relation with other properties
- Subgroup in which every relatively normal subgroup is strongly closed
- Subgroup in which every relatively normal subgroup is weakly closed
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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Any CEP subgroup of a CEP subgroup is a CEP subgroup. This follows from the fact that the property of being CEP is a balanced subgroup property with respect to a suitable formalism.
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The trivial subgroup is also a CEP subgroup, so the property of being a CEP subgroup is trivially true.
Thus, the property of being a CEP subgroup is a trim subgroup property.