Index-unique subgroup

Definition
A defining ingredient::subgroup of finite index in a group is termed index-unique if it is the only subgroup of that particular index in the whole group.

Stronger properties

 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup

Weaker properties

 * Stronger than::Quotient-isomorph-free subgroup
 * Stronger than::Characteristic subgroup

Related properties
For a finite group, an index-unique subgroup is the same thing as an order-unique subgroup. This follows from Lagrange's theorem.

Facts
If the index of the commutator subgroup is a prime number, or the square of a prime number, or, more generally, any Abelianness-forcing number, then the commutator subgroup is index-unique.