Let be a Group homomorphism, and write for the set . Then is a group under the operation inherited from .
Proof
To prove this, we must verify the group axioms. Let be a group homomorphism, and let be the identities of and of respectively. Write for the image of .
Then is closed under the operation of : since , so the result of -multiplying two elements of is also in .
is the identity for : it is , so it does lie in the image, while it acts as the identity because , and likewise for multiplication on the right.
Inverses exist, by "the inverse of the image is the image of the inverse".
The operation remains associative: this is inherited from .
Therefore, is a group, and indeed is a subgroup of .