Group homomorphism

In mathematics, a group homomorphism is a special kind of function that connects two groups. Groups are sets of elements with a rule for combining them, like adding or multiplying numbers. A group homomorphism takes an element from one group and maps it to another group so that the rule for combining elements still works.

If we have two groups, (G, ∗) and (H, ·), a group homomorphism from G to H ensures that when we combine any two elements in G using ∗ and then apply the homomorphism, it is the same as applying the homomorphism to each element first and then combining them in H using ·.

One important property of a group homomorphism is that it maps the identity element of G to the identity element of H. The identity element is a special element that, when combined with any other element, leaves it unchanged. Additionally, a homomorphism maps inverses to inverses. An inverse of an element is another element that, when combined with the original, results in the identity element. These properties ensure that the homomorphism works well with the group structure.

In more advanced areas of mathematics, such as with topological groups, a homomorphism may also need to preserve extra structure. For example, a homomorphism of topological groups is often required to be continuous, meaning it preserves the way elements are close to each other. This makes group homomorphisms useful for studying the relationships between different mathematical structures.

Properties

A group homomorphism is a special kind of function between two groups. It makes sure the groups' rules still work. If you take any two elements from the first group, combine them, and then use the function, it's the same as using the function on each element first and then combining.

Because of this rule, the function sends the special "identity" element from the first group to the identity element of the second group. It also sends each element's "inverse" to the inverse in the second group. This keeps the structure of both groups working together.

Types

There are special kinds of group homomorphisms.

A monomorphism is a group homomorphism that is injective. This means it keeps things separate and distinct.

An epimorphism is a group homomorphism that is surjective. This means it covers every point in the group it is mapping to.

An isomorphism is a group homomorphism that is bijective. This means it is both injective and surjective. When two groups are isomorphic, they work the same way, even if their elements have different names.

An endomorphism is a group homomorphism where the starting group and the ending group are the same.

An automorphism is a bijective endomorphism. All the automorphisms of a group form another group called the automorphism group.

For example, the automorphism group of (Z, +) has only two elements: the identity transformation and multiplication by −1; it is isomorphic to (Z/2Z, +).

Main article: Monomorphism

Main article: Epimorphism

Main article: Isomorphism

Main article: Endomorphism

Main article: Automorphism

Image and kernel

Main articles: Image (mathematics) and kernel (algebra)

In group homomorphisms, we learn about two key ideas: the kernel and the image.

The kernel of a homomorphism h is the group of elements in the first group G that h sends to the identity element in the second group H. This shows us how h works.

The image of h is all the elements in H that come from using h on elements of G. The kernel and image together help us understand how well h connects the two groups.

Examples

Here are a few simple examples of group homomorphisms:

• Imagine a small group with three elements, like a clock that only shows three hours. There is a bigger group of all whole numbers. A special rule can match each whole number to a position on the small clock, keeping the group rules the same.
• The exponential map connects adding real numbers to multiplying positive real numbers, keeping the group rules.
• The same idea works with complex numbers, linking addition to multiplication of non-zero complex numbers.

Category of groups

If h : G → H and k : H → K are group homomorphisms, then putting them together makes another homomorphism k ∘ h : G → K. This shows that all groups and the special maps between them, called homomorphisms, work together to form a structure called the category of groups.

Homomorphisms of abelian groups

When groups are commutative, or abelian, all the homomorphisms between two such groups form another abelian group. We can add two homomorphisms by adding what they give for each element in the group. This works because the group we are mapping to is also commutative.

This addition of homomorphisms works well with combining homomorphisms together. If we have a homomorphism from one group to another and then add two homomorphisms from that group to a third group, the result is the same as adding first and then combining. This property helps show that the set of all homomorphisms from a group to itself forms a special algebraic structure called a ring. For instance, the endomorphism ring of certain abelian groups can be represented by matrices.