T*-group

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 defining ingredient::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 defining ingredient::intermediately subnormal-to-normal subgroup.

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

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Dedekind group
 * Weaker than::Group in which every subgroup is pronormal

Weaker properties

 * Stronger than::T-group
 * Stronger than::HN*-group
 * Stronger than::HN-group

Metaproperties
Any subgroup of a T*-group is a T*-group. This follows from the definition.