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