Group homomorphism

In mathematics, a group homomorphism is a special kind of function that connects two groups while preserving their group properties. 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 in a way 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 ·. This means the homomorphism respects the way elements interact within the groups.

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 "is compatible 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 in a topological sense. This makes group homomorphisms a powerful tool for studying the relationships between different mathematical structures.

Properties

A group homomorphism is a special kind of function between two groups that keeps the group's rules working properly. This means if you take any two elements from the first group, combine them using the group's operation, and then apply the function, it's the same as applying the function to each element first and then combining those results in the second group.

Because of this rule, the function sends the special "identity" element — the one that doesn't change other elements when combined with them — from the first group to the identity element of the second group. It also sends each element's "inverse" — another element that combines with it to produce the identity — to the inverse of the image of that element in the second group. This makes the function compatible with the structure of both groups.

Types

There are special kinds of group homomorphisms. A monomorphism is a group homomorphism that is injective, meaning it preserves distinctness. An epimorphism is a group homomorphism that is surjective, meaning it reaches every point in the target group.

An isomorphism is a group homomorphism that is bijective, meaning it is both injective and surjective. When two groups are isomorphic, they are identical for all practical purposes, except in how their elements are labeled. An endomorphism is a group homomorphism where the domain and codomain are the same group. An automorphism is a bijective endomorphism, and the collection of all automorphisms of a group forms 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 look at two important ideas: the kernel and the image. The kernel of a homomorphism h is the set of elements in the first group G that h maps to the identity element in the second group H. This helps us understand how h behaves.

The image of h is the set of all elements in H that come from applying h to elements of G. Together, the kernel and image help us see how close h is to being a perfect match between the two groups.

Examples

Here are a few simple examples of group homomorphisms:

• Imagine a group with three elements, like a clock that only has three hours. There's also 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 intact.
• 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 combining them creates another homomorphism k ∘ h : G → K. This means that all groups and the special maps between them called homomorphisms fit together to form what mathematicians call a category, known as the category of groups.

Homomorphisms of abelian groups

When groups are commutative, or abelian, the collection of all homomorphisms between two such groups forms another abelian group. We can add two homomorphisms by adding their results 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, 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.