# Prime index and quotient-subisomorph-containing implies index-unique

## Statement

Suppose is a group, and is a Subgroup of prime index (?) (specifically, a Normal subgroup of prime index (?)) that is also a Quotient-subisomorph-containing subgroup (?). Then, is an Index-unique subgroup (?): there is no other subgroup of with the same index.

When is a finite group, this is equivalent to saying that is an order-unique subgroup: there is no other subgroup of of the same order.

## Related facts

## Proof

**Given**: A group , a prime number , a subgroup of prime index in such that is contained in the kernel of any homomorphism from to .

**To prove**: is the only subgroup of index in .

**Proof**: Suppose is a subgroup of index . Then, . Thus, the quotient map can be composed with this isomorphism, giving a map with kernel . Since is quotient-homomorph-containing in , this implies . But since both have index , this forces .