Index-unique subgroup

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Definition

A 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.

Relation with other properties

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup

Weaker properties

• Quotient-isomorph-free subgroup
• 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.