Kernel of a group homomorphism

Given a Group homomorphism , with identity , the kernel is defined as #m/def/group

Every kernel is a Normal subgroup, #m/thm/group and vice versa (see quotient group).

Proof of normal subgroup

Clearly . Let , then . It follows , thus . Therefore is a subgroup by One step subgroup test. Now let . Then for any , , whence . Therefore is a Normal subgroup.

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