CEP-subgroup

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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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Definition

Symbol-free definition

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

A subgroup H of a group G is termed a CEP subgroup if for any normal subgroup N of H, there is a normal subgroup M of G such that N = MH.

Formalisms

In terms of the subgroup intersection extension formalism

In terms of the subgroup intersection extension formalism, the property of being a CEP subgroup is the balanced property with respect to the subgroup property of normality.

Facts

Every subgroup of a group is a CEP-subgroup if and only if the group is a solvable T-group.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Transitivity

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.

Trimness

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 improper subgroup, or the whole group, is clearly a CEP subgroup, so the property of being a CEP subgroup is identity-true.

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.