Talk:Linear group

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I have listed below some ambitious goals for this page but don't have the mathematical foundation to realize them. Any help would be greatly appreciated.

First it seems that some of the definitions of the classical groups only consider the case where the field is R or C. I've left some comments on the talk pages for these groups so hopefully someone there will generalize the definitions.

Second it would be nice to provide short definitions for each of the types of classical groups listed on the page.

Third I believe the list can be increased to include "combinations" of these groups, such as the special orthogonal group and projective special orthogonal group. Unfortunately I'm not even sure which definitions make sense -- for instance I believe the symplectic group has a trivial center so it wouldn't make sense to talk about the projective symplectic group, but I'm not really sure about this. In short if someone could verify which of these group definitions are meaningful it would be very helpful.

Classical Groups

It would be nice to give short definitions of these groups and specify when they are simple (IE for which n and F)

• GL(n,F) - group is well covered in general linear group
• U(n,F) - definition of unitary group only uses C (left request on talk page)
• Sp(n,F) - definition of symplectic matrix appears general enough to cover arbitrary fields
• O(n,F) - The orthogonal group page gives a definition over arbitrary fields. Could use some examples over finite fields.

Special Linear Groups

When are these groups simple? When is the center nontrivial?

• SL(n,F) - group is well covered in general linear group
• SU(n,F) - definition of special unitary group only uses C (left request on talk page)
• SSp(n,F) - Sp is already a subgroup of SL so SSp = Sp
• SO(n,F) - The orthogonal group page gives a definition over arbitrary fields. Could use some examples over finite fields.

Projective Groups

When are these groups simple?

• PGL(n,F) - group is covered in projective linear group. Definition could be made more clear on how PGL is a matrix group.
• PU(n,F) - definition of projective unitary group only uses C
• PSp(n,F) - projective symplectic group needs to be written
• PO(n,F) - projective orthogonal group needs to be written

Projective Special Linear Groups

When are these groups simple?

• PSL(n,F) - group is covered in projective linear group. Definition could be made more clear on how PGL is a matrix group.
• PSU(n,F) - projective special unitary group needs to be written
• PSSp(n,F) - As mentioned above Sp is a subgroup of SL
• PSO(n,F) - projective special orthogonal group needs to be written

Contents 1 question 2 Representation theory of finite groups 3 Italic group names? 4 Misleading

[edit] question

what's meant by rare in " GO_n(R) (rare) "? thanks. Mct mht 05:53, 22 April 2006 (UTC)

"uncommon". I deleted it since it seems to be confusing. R.e.b. 14:29, 22 April 2006 (UTC)

[edit] Representation theory of finite groups

Nice to see a new section on finite groups as matricies, I instantly though of Representation theory of finite groups which seems to be related. Should this be worked into the article? --Salix alba (talk) 20:25, 26 April 2006 (UTC)

Good idea. I just added a section on representation theory and character theory which refers the reader to the relevant sections. TooMuchMath 02:47, 27 April 2006 (UTC)

[edit] Italic group names?

In this article, and in those to which it links, it would be nice to be uniform in our notation. Which do we prefer?:

1. GL_n(R)
2. GL_n(R)
3. GL_n(R)
4. something completely different

Here I've adopted the first option, GL_n(R), but the second seems more readable. At least this article is now self-consistent, which is more than I can say for others. --KSmrq^T 19:12, 27 April 2006 (UTC)

I think it looks pretty good as it is. The full italics causes the term to stand out from the rest of the text. TooMuchMath 20:04, 27 April 2006 (UTC)

[edit] Misleading

This article is very misleading. While classical groups are very important, they neither exhaust nor are even typical as linear/matrix groups.

• Linear groups do not have to be algebraic.
• Linear groups do not have to be (semi)simple. For example, Heisenberg group is a fairly typical example of a linear group, but it's nilpotent.
• Since all finite groups are already permutation groups, establishing their linearity (over Z or any field) does not add much to their theory. On the other hand, among infinite groups linear groups may form the class that is the simplest to understand, but generally speaking, they are neither concrete, nor the most common (and one would have to be pressed hard to justify what is meant by common for infinite groups).

I think it would be good to talk about some general properties of finitely generated linear groups, such as residual finiteness, and lead to examples of non-linear groups, but I feel that this is rather divergent from the current state of the page. Arcfrk 06:56, 31 March 2007 (UTC)

I assume (hope?) this is an article in transition. Recently a proposal was made to merge "matrix group" with Lie groups, and I immediately protested that that was a logical impossibility. So instead it was merged here. Personally, I have no attachment to the present contents, and encourage you to do what you think best. --KSmrq^T 21:06, 31 March 2007 (UTC)