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

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Discussion Overview

The discussion revolves around the classification of homomorphisms from abstract groups to the general linear group GL(n). Participants explore the relevant fields of study and the implications of these homomorphisms in various mathematical contexts, including abstract algebra and representation theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants suggest that abstract algebra encompasses the study of homomorphisms of groups, rings, fields, and more, but do not identify a specific term for classifying homomorphisms of groups.
  • One participant lists several fields that may relate to the classification of homomorphisms, including category theory, group theory, and representation theory, while noting the importance of specifying which groups are being considered.
  • Another participant emphasizes the significance of GL(n) and states that the study of homomorphisms from abstract groups to GL(n) is referred to as (linear) representation theory of groups.
  • A reference to a classic text by Serre on the subject of finite groups is provided, indicating a resource for further exploration.

Areas of Agreement / Disagreement

Participants express varying perspectives on the classification of homomorphisms, with no consensus on a single term or field of study that encapsulates the discussion. Multiple viewpoints regarding the relevance of different mathematical areas remain present.

Contextual Notes

The discussion lacks clarity on specific types of groups being referenced, which may influence the classification of homomorphisms. Additionally, the implications of these homomorphisms in different mathematical contexts are not fully resolved.

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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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