Group homomorphism

A group homomorphism is a morphism in $𝖦 𝗋 𝗉$ , that is to say it is a structure-preserving map between groups. #m/def/group Let $(𝐺, \circ)$ and $(𝐻, •)$ be groups, and let $𝑓 : 𝐺 \to 𝐻$ . Then $𝑓$ is a homomorphism iff for any $𝑎, 𝑏 \in 𝐺$

𝑓 (𝑎 \circ 𝑏) = 𝑓 (𝑎) • 𝑓 (𝑏)

It immediately follows that $𝑓 (𝑒) = 𝑒$ and $𝑓 (𝑎 - 1) = 𝑓 (𝑎) - 1$ .

Proof

For the identity property, it is clear that $𝑓 (𝑎 \circ 𝑒) = 𝑓 (𝑎) = 𝑓 (𝑎) • 𝑓 (𝑒)$ for any $𝑎 \in 𝐺$ , hence $𝑓 (𝑒) = 𝑒$ . For the latter property, notice that for any $𝑎 \in 𝐺$ it follows $𝑓 (𝑎 \circ 𝑎 - 1) = 𝑓 (𝑎) • 𝑓 (𝑎 - 1) = 𝑓 (𝑒) = 𝑒$ , so $𝑓 (𝑎 - 1) = 𝑓 (𝑎) - 1$ .

A bijective homomorphism is the a group isomorphism. Isomorphic groups have the same group table, and are essentially the same up to relabelling.

Group monomorphism, Group epimorphism
The Kernel of a group homomorphism $k e r (𝑓)$ is the set of all domain elements that map to the identity, and it forms a normal subgroup (proof in Zettel)
The image $𝑓 (𝐺)$ is the range of $𝑓$ , and The image of a group homomorphism is a subgroup.
A group homomorphism induces a subgroup homomorphism when its domain is restricted.

#state/tidy | #lang/en | #SemBr

Group homomorphism

Properties and related