Talk:Linear group

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)

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?: Here I've adopted the first option, GLn(R), but the second seems more readable. At least this article is now self-consistent, which is more than I can say for others. --KSmrqT 19:12, 27 April 2006 (UTC)
 * 1) GLn(R)
 * 2) GLn(R)
 * 3) $$GL_n(R)$$
 * 4) something completely different


 * 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)

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. --KSmrqT 21:06, 31 March 2007 (UTC)

PGL
Why should PGL be a matrix group? By, definition, it is not, since its elements are not matrices. I don't think it's even a linear group in general... --Roentgenium111 (talk) 18:25, 27 January 2012 (UTC)

Infinitely generated non-example
As noted in the justification for the citation-needed talk it is not obvious that the group $$(\mathbb Z/2\mathbb Z)^{\mathbb N}$$ is non-linear. In fact it is false: it is linear over any infinite field of characteristic 2 (since it is a subgroup of the additive group which is linear). On the other hand it cannot be linear in any other characteristic, so adding a factor $$(\mathbb Z/3\mathbb Z)^{\mathbb N}$$ does indeed give a non-linear group (in any characteristic), which maybe would be better written as $$(\mathbb Z/6\mathbb Z)^{\mathbb N}$$. The proof is elementary (one of the two factors must be conjugated to a diagonal group, and since all finite-exponent subgroups of the multiplicative group of a field are finite this is impossible). I think it would not be OR if written down in the article. jraimbau (talk) 07:21, 24 June 2021 (UTC)