Talk:Group theory: Difference between revisions
imported>Jared Grubb (→A few thoughts: a brainstorming session!) |
imported>Catherine Woodgold (Definitions needed) |
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Great suggestions! I've created [[{{FULLPAGENAME}}/Brainstorming]] so we can brainstorm about this topic: what needs to be here, what should be at [[Group (mathematics)]]. - [[User:Jared Grubb|Jared Grubb]] 10:40, 4 May 2007 (CDT) | Great suggestions! I've created [[{{FULLPAGENAME}}/Brainstorming]] so we can brainstorm about this topic: what needs to be here, what should be at [[Group (mathematics)]]. - [[User:Jared Grubb|Jared Grubb]] 10:40, 4 May 2007 (CDT) | ||
== Definitions needed == | |||
Interesting topic. So group theory is how it was proven that quintics can't be solved. (If I knew that before I'd forgotten it.) | |||
This is where you lose me: ''A solvable group, or a soluble group, is a group that has a normal series whose quotient groups are all abelian. '' Definitions are needed for: "normal series", "quotient group", "alternating subgroup", "<math>A_5</math>", "symmetric group". | |||
In this sentence: ''A free group is a group in which every element of the group is a unique product, or string, of elements of some subset of the group.'' It needs to be clarified whether "unique" means that each element can be expressed as a product in only one way (up to use of the identity element, presumably), or whether it means that a given string can only represent one element (obviously true given the definition of binary operation) or perhaps that a given subset can only represent one element regardless of which order they're put into a string. --[[User:Catherine Woodgold|Catherine Woodgold]] 20:25, 5 May 2007 (CDT) |
Revision as of 20:25, 5 May 2007
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. ), Lie groups, or much more complex groups. Some obvious examples of finite groups are:
- (finite) cyclic groups
- direct sums of cyclic groups
- the symmetric groups and alternatiing groups
- the dihedral groups
- the unit quaternions
Beyond that, there are the "classical" groups which are the analogues of linear Lie groups over finite fields (e.g., and .
This article should also talk about representations of groups (i.e., homomorphisms ), 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:
- group actions
- group presentations by generators and relations
- the isomorphism theorems
- the "Burnside" lemma (which is not due to Burnside, but the name is traditional)
- the Sylow theorems
- applications to Galois theory
- 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)
Great suggestions! I've created Talk:Group theory/Brainstorming so we can brainstorm about this topic: what needs to be here, what should be at Group (mathematics). - Jared Grubb 10:40, 4 May 2007 (CDT)
Definitions needed
Interesting topic. So group theory is how it was proven that quintics can't be solved. (If I knew that before I'd forgotten it.)
This is where you lose me: A solvable group, or a soluble group, is a group that has a normal series whose quotient groups are all abelian. Definitions are needed for: "normal series", "quotient group", "alternating subgroup", "", "symmetric group".
In this sentence: A free group is a group in which every element of the group is a unique product, or string, of elements of some subset of the group. It needs to be clarified whether "unique" means that each element can be expressed as a product in only one way (up to use of the identity element, presumably), or whether it means that a given string can only represent one element (obviously true given the definition of binary operation) or perhaps that a given subset can only represent one element regardless of which order they're put into a string. --Catherine Woodgold 20:25, 5 May 2007 (CDT)
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