What Is the Study of Homomorphisms from Abstract Groups to GL(n)?

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In summary: Summary: Category theory, abstract algebra, and various other fields such as group theory, commutative algebra, Galois theory, homological algebra, and Lie theory all involve the study of homomorphisms of groups. These homomorphisms are important for comparing different groups, particularly abstract groups to the group GL(n) of automorphisms of an n-dimensional vector space. This subject, known as (linear) representation theory of groups, is well understood and has been extensively studied, with classic references such as Serre's book on finite groups.
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What is the field of study called that classifies homomophisms of groups?
What is the field of study called that classifies homomophisms of groups?
 
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Abstract algebra covers homomorphisms of groups, rings, fields, modules, vector spaces, and more. I do not know of a specific name that answers your exact question.

What are you actually trying to accomplish, maybe finding a reference or text?
 
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zwoodrow said:
Summary: What is the field of study called that classifies homomophisms of groups?

What is the field of study called that classifies homomophisms of groups?
Category theory, abstract algebra, group theory, commutative algebra, Galois theory, homological algebra, Lie theory, and probably some more, e.g. crystallography. As soon as one considers groups, as soon are homomorphisms involved. The missing information is: Which groups?
 
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one very important and well understood group is GL(n), automorphisms of an n dimensional vector space. the usefulness of homomorphisms is that they allow us to compare different groups. it is of some importance to compare abstract groups to GL(n). this subject, the study of homomorphisms from abstract groups to GL(n) is called (linear) representation theory (of groups). here is a classic reference by serre, in the case of finite groups:

https://www.alibris.com/search/books/isbn/9780387901909?invid=17220358343&utm_source=Google&utm_medium=cpc&utm_campaign=NMPi&gclid=CjwKCAjwx7GYBhB7EiwA0d8oe9SL-wf5H1xmf2dgYNV9LEvH9sWaIjmttAPzAycg1pvh21IMrqZkcBoC6jAQAvD_BwE&gclsrc=aw.ds
 
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FAQ: What Is the Study of Homomorphisms from Abstract Groups to GL(n)?

1. What is the purpose of classifying group homomorphisms?

The purpose of classifying group homomorphisms is to better understand the structure and properties of groups. By studying the different types of homomorphisms between groups, we can gain insights into the relationships and similarities between different groups.

2. How do you define a group homomorphism?

A group homomorphism is a function that preserves the algebraic structure of a group. In other words, it maps elements from one group to another in such a way that the group operation is preserved. This means that the result of applying the group operation to two elements in the first group will be the same as the result of applying the homomorphism to those elements and then applying the group operation in the second group.

3. What are the different types of group homomorphisms?

There are several types of group homomorphisms, including monomorphisms, epimorphisms, and isomorphisms. Monomorphisms are injective homomorphisms, meaning they preserve distinctness. Epimorphisms are surjective homomorphisms, meaning they cover all elements of the target group. Isomorphisms are bijective homomorphisms, meaning they are both injective and surjective, and therefore preserve both distinctness and coverage.

4. How do you classify group homomorphisms?

Group homomorphisms can be classified based on their properties, such as injectivity, surjectivity, and bijectivity. They can also be classified based on the type of group they map from and the type of group they map to. For example, a homomorphism from a cyclic group to a non-cyclic group would be classified differently than a homomorphism from a non-cyclic group to a cyclic group.

5. What are some real-world applications of classifying group homomorphisms?

Classifying group homomorphisms has many practical applications, such as in cryptography, coding theory, and data compression. For example, in cryptography, group homomorphisms can be used to encrypt and decrypt data, as well as to verify the integrity of the data. In coding theory, group homomorphisms can be used to encode and decode data, while in data compression, they can be used to reduce the size of data without losing important information.

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