# Index-unique subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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

### Weaker properties

### 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.