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| It simply is not true that the translation group is "our first example of a ]" (as the article is currently arranged). (<b>Z</b>,+) is also a Lie group; it is simply ''discrete'', or 0 dimensional. 0 dimensional Lie groups (discrete groups) are studied in ordinary group theory rather than Lie theory, but they are still technically Lie groups. Hence the necessity for the adjective "nondiscrete". (If you want to change "nondiscrete" to "nontrivial", then I won't fight that, although I won't advocate it either.) -- ], 2002/04/03 | It simply is not true that the translation group is "our first example of a ]" (as the article is currently arranged). (<b>Z</b>,+) is also a Lie group; it is simply ''discrete'', or 0 dimensional. 0 dimensional Lie groups (discrete groups) are studied in ordinary group theory rather than Lie theory, but they are still technically Lie groups. Hence the necessity for the adjective "nondiscrete". (If you want to change "nondiscrete" to "nontrivial", then I won't fight that, although I won't advocate it either.) -- ], 2002/04/03 | ||
| Is it common to allow 0-dimensional manifolds? What could possibly be gained by that? Every set is a 0-dimensional manifold. In EDM they don't specify what ''n'' is in an ''n''-dimensional manifold, but from their definition of "manifold with boundary" it is clear that they implicitly assume ''n''>0. Furthermore, would you typically find '''Z'''<sub>''p''</sub> in a list of simple Lie groups? | Is it common to allow 0-dimensional manifolds? What could possibly be gained by that? Every set is a 0-dimensional manifold. In EDM they don't specify what ''n'' is in an ''n''-dimensional manifold, but from their definition of "manifold with boundary" it is clear that they implicitly assume ''n''>0. Furthermore, would you typically find or expect '''Z'''<sub>''p''</sub> in a list of simple Lie groups? | ||
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Revision as of 09:47, 4 April 2002
We currently have two different group pages: Mathematical group and Mathematical Group. I suggest simply deleting Mathematical group and redirecting it to Mathematical Group.
- Seconded, with one caveat: the title of the final article should be "Mathematical group" to comply with naming standards. --AxelBoldt
Done.
Zundark, 2001-08-11
The axiom of closure:
(Closure) for all a and b in G, a * b belong to G.
is superfluous, by definition of a binary operation. It's worth mentioning that closure follows from the definition, though.
The test of closure in the examples is in fact a test that the described mapping is inded a binary operation.
Any thoughts before I wade on in and make changes?
"This was our first example of a non-abelian group, because the operation o here is not commutative as the table shows. " If the table did show commutativity, would it be symmetrical about the diagonal from top left to bottom right? TimJ 5 Feb 2002
- Yes. The group is abelian if and only if the table is symmetric about the main diagonal. --Zundark, 2002 Feb 5
Could someone put up a good description of Sylow's Theorem?
- See Sylow theorems.
Also, it'd be nice to see a page dedicated to examples of groups.
Nice exercise: Classify all (isomorphism classes of) groups of order <=60. I'd like to see a page on that.
- Do you realise how complicated this is, especially for order 32? Doing order <= 15 might be feasible, however. --Zundark, 2002 Feb 22
It simply is not true that the translation group is "our first example of a Lie group" (as the article is currently arranged). (Z,+) is also a Lie group; it is simply discrete, or 0 dimensional. 0 dimensional Lie groups (discrete groups) are studied in ordinary group theory rather than Lie theory, but they are still technically Lie groups. Hence the necessity for the adjective "nondiscrete". (If you want to change "nondiscrete" to "nontrivial", then I won't fight that, although I won't advocate it either.) -- Toby, 2002/04/03
Is it common to allow 0-dimensional manifolds? What could possibly be gained by that? Every set is a 0-dimensional manifold. In EDM they don't specify what n is in an n-dimensional manifold, but from their definition of "manifold with boundary" it is clear that they implicitly assume n>0. Furthermore, would you typically find or expect Zp in a list of simple Lie groups? AxelBoldt