# 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

Children:

- Kernel of group homomorphism
- Image of the identity under a group homomorphism is the identity
All group homomorphisms preserve the identity.

- Under a group homomorphism, the image of the inverse is the inverse of the image
The operations of “taking inverses” and “applying a group homomorphism” commute: it does not matter in which order we do them.

- The image of a group under a homomorphism is a subgroup of the codomain
Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.

- The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.

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:

Thing. Implications implications implications. (Proof.)

Thing. Implications implications. (Proof.)

…

I’d then eventually add an intro lens.”

Having proofs on child pages makes sense to me too.