# T*-group

From Groupprops

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to T*-group, all facts related to T*-group) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

## 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 is said to be a T*-group if whenever , we have . In other words, any subgroup of 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

View a complete list of subgroup-closed group properties

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

View a complete list of quotient-closed group properties