T*-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 group is said to be a T*-group if it satisfies the following equivalent conditions:

Every subgroup of it is a T-group
If a subgroup is subnormal in some intermediate subgroup, it is also normal in that intermediate subgroup.
Every subgroup of it is an intermediately subnormal-to-normal subgroup.

Definition with symbols

A group $G$ is said to be a T*-group if whenever $H\triangleleft K\triangleleft N\leq G$ , we have $H\triangleleft N$ . In other words, any subgroup $N$ of $G$ is a T-group.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Any subgroup of a T*-group is a T*-group. This follows from the definition.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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