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> Group theory originated in a very visual and geometric ways studying the symmetries of various objects Did it? As far as I know, group theory started as meth

by seppel 9y ago

> Group theory originated in a very visual and geometric ways studying the symmetries of various objects Did it? As far as I know, group theory started as method to describe the possible permutations of roots of polynomials, so it was very abstract right from the beginning. And while there are certain geometric interpretation of some groups and the ways these groups act on objects, the "visual" part stops as soon as the groups gets more general or have more structure.

zornthewise 9y ago
You are correct, group theory did start with the permutations of roots of polynomials. In the other hand, the modern perspective is to consider these groups as coming from geometry (the etale fundamental group point of view) and is really a very successful theory. I think it is generally useful to try and give a geometric interpretation to whatever we are interested in and groups certainly benefit from such an approach. Moreover, the groups relevant to physics are very geometric things (they are literally spaces in the sense that the real number line is a space ) and the geometry plays a large role in these groups. In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play. Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
- seppel 9y ago
  > In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play. Number theory is another example, finding roots of polynomials was already mentioned. > Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory). It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)

by seppel 9y ago

zornthewise 9y ago

You are correct, group theory did start with the permutations of roots of polynomials. In the other hand, the modern perspective is to consider these groups as coming from geometry (the etale fundamental group point of view) and is really a very successful theory. I think it is generally useful to try and give a geometric interpretation to whatever we are interested in and groups certainly benefit from such an approach. Moreover, the groups relevant to physics are very geometric things (they are literally spaces in the sense that the real number line is a space ) and the geometry plays a large role in these groups. In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play. Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).

seppel 9y ago
> In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play. Number theory is another example, finding roots of polynomials was already mentioned. > Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory). It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)