# Group homomorphism

A group homomorphism is a function between groups which “respects the group structure”.

# Definition

Formally, given two groups $$(G, +)$$ and $$(H, *)$$ (which hereafter we will abbreviate as $$G$$ and $$H$$ respectively), a group homomorphism from $$G$$ to $$H$$ is a function $$f$$ from the underlying set $$G$$ to the underlying set $$H$$, such that $$f(a) \* f(b) = f(a+b)$$ for all $$a, b \in G$$.

# Examples

• For any group $$G$$, there is a group homomorphism $$1_G: G \to G$$, given by $$1_G(g) = g$$ for all $$g \in G$$. This homomorphism is always bijective.

• For any group $$G$$, there is a (unique) group homomorphism into the group $$\{ e \}$$ with one element and the only possible group operation $$e \* e = e$$. This homomorphism is given by $$g \mapsto e$$ for all $$g \in G$$. This homomorphism is usually not injective: it is injective if and only if $$G$$ is the group with one element. (Uniqueness is guaranteed because there is only one function, let alone group homomorphism, from any set $$X$$ to a set with one element.)

• For any group $$G$$, there is a (unique) group homomorphism from the group with one element into $$G$$, given by $$e \mapsto e_G$$, the identity of $$G$$. This homomorphism is usually not surjective: it is surjective if and only if $$G$$ is the group with one element. (Uniqueness is guaranteed this time by the property proved below that the identity gets mapped to the identity.)

• For any group $$(G, +)$$, there is a bijective group homomorphism to another group $$G^{\mathrm{op}}$$ given by taking inverses: $$g \mapsto g^{-1}$$. The group $$G^{\mathrm{op}}$$ is defined to have underlying set equal to that of $$G$$, and group operation $$g +_{\mathrm{op}} h := h + g$$.

• For any pair of groups $$G, H$$, there is a homomorphism between $$G$$ and $$H$$ given by $$g \mapsto e_H$$.

• There is only one homomorphism between the group $$C_2 = \{ e_{C_2}, g \}$$ with two elements and the group $$C_3 = \{e_{C_3}, h, h^2 \}$$ with three elements; it is given by $$e_{C_2} \mapsto e_{C_3}, g \mapsto e_{C_3}$$. For example, the function $$f: C_2 \to C_3$$ given by $$e_{C_2} \mapsto e_{C_3}, g \mapsto h$$ is not a group homomorphism, because if it were, then $$e_{C_3} = f(e_{C_2}) = f(gg) = f(g) f(g) = h h = h^2$$, which is not true. (We have used that the identity gets mapped to the identity.)

# Properties

• The identity gets mapped to the identity. (Proof.)

• The inverse of the image is the image of the inverse. (Proof.)

• The image of a group under a homomorphism is another group. (Proof.)

• The composition of two homomorphisms is a homomorphism. (Proof.)

Children:

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

• I have a question about general Arbital practice here. A mathematician will probably already know what a group homomorphism is, but they probably also don’t need the proofs of the Properties, for instance, and they don’t need the explanation of the trivial group. Should I have split this up into different lenses in some way?

• so8res: “I would set up the page as follows:

A group homomorphism is X. Key properties of group homomorphisms include:

1. Thing. Implications implications implications. (Proof.)

2. Thing. Implications implications. (Proof.)

I’d then eventually add an intro lens.”

Having proofs on child pages makes sense to me too.