Talk:Group theory

Article Checklist for "Group theory"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Stub: no more than a few sentences
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Jared Grubb 15:43, 3 May 2007 (CDT)

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I have written quite a bit on groups, and it would be nice to have someone help make it more readable. I think the "examples" section looks a bit daunting to the eye, but I'm not sure how to organize it any better. - Jared Grubb 23:59, 3 May 2007 (CDT)

A few thoughts

It's worth noting that groups can be roughly divided into finite and infinite groups. The infinite groups may be discrete groups closely related to the finite ones (e.g. $\mathbb {Z}$), Lie groups, or much more complex groups. Some obvious examples of finite groups are:

1. (finite) cyclic groups
2. direct sums of cyclic groups
3. the symmetric groups and alternatiing groups
4. the dihedral groups
5. the unit quaternions

Beyond that, there are the "classical" groups which are the analogues of linear Lie groups over finite fields (e.g., $SL(n,\mathbb {F} _{q})$ and $PSL(n,\mathbb {F} _{q})$.

This article should also talk about representations of groups (i.e., homomorphisms $\rho :G\rightarrow GL(n,\mathbb {C} )$), and this would be an excellent place to mention that there are exactly 5 regular polyhedra. The complete classification of finite simple groups needs to be mentioned, too.

Other topics from group theory should probably include

1. group actions
2. group presentations by generators and relations
3. the isomorphism theorems
4. the "Burnside" lemma (which is not due to Burnside, but the name is traditional)
5. the Sylow theorems
6. applications to Galois theory
7. Klein's Erlangen program (characterization of geometries in terms of the group of symmetries of the geometry)

It might be reasonable to talk about applications of group theory to classical and quantum mechanics, too. Greg Woodhouse 04:40, 4 May 2007 (CDT)