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

[zwoodrow]

TL;DR

What is the field of study called that classifies homomophisms of groups?

What is the field of study called that classifies homomophisms of groups?

 

[jim mcnamara]

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?

 

[fresh_42]

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?

 

[mathwonk]

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