Group homomorphism

Kernel of a group homomorphism

Given a Group homomorphism 𝑓 :𝐺 →𝐻, with identity 𝑒𝐻 ∈𝐻, the kernel ker⁡𝑓 is defined as group

ker⁡𝑓={𝑔∈𝐺:𝑓(𝑔)=𝑒𝐻}

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

Proof of normal subgroup

Clearly 𝑒 ∈ker⁡𝑓. Let 𝑎,𝑏 ∈ker⁡𝑓, then 𝑓(𝑎) =𝑓(𝑏) =𝑒. It follows 𝑓(𝑎𝑏−1) =𝑓(𝑎)𝑓(𝑏−1) =𝑓(𝑎)𝑓(𝑏)−1 =𝑒, thus 𝑎𝑏−1 ∈ker⁡𝑓. Therefore ker⁡𝑓 is a subgroup by One step subgroup test. Now let 𝑘 ∈ker⁡𝑓. Then for any 𝑔 ∈𝐺, 𝑓(𝑔𝑘𝑔−1) =𝑓(𝑔)𝑓(𝑘)𝑓(𝑔−1) =𝑓(𝑔)𝑒𝑓(𝑔−1) =𝑓(𝑒) =𝑒, whence 𝑔𝑘𝑔−1 ∈ker⁡𝑓. Therefore ker⁡𝑓 is a Normal subgroup.

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