The image of a group under a homomorphism is a subgroup of the codomain

https://arbital.com/p/image_of_group_under_homomorphism_is_subgroup

by Patrick Stevens Jun 14 2016

Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.


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 .