Talk:Projective linear group

Notation Error?[edit]

It seems to me that in the section "Topology", in the paragraph on PGL(n,C), there are two variables called "n" - one numbering the homotopy groups, and one numbering the dimension in PGL(n,C). This makes the actual statement unclear. I don't feel qualified to fix this myself without introducing other errors. 2003:DF:2711:13C8:F4CE:92BE:B665:FBB2 (talk) 13:53, 5 April 2021 (UTC)[reply]

Simple?[edit]

Section added. —Nils von Barth (nbarth) (talk) 00:16, 2 November 2009 (UTC)[reply]

As I remembered, PSL(n,K ) was simple for any field, as long as n>=3 or K has at least four elements. I thought you didn't have to assume K to be a Galois field? Evilbu 19:20, 10 February 2006 (UTC)[reply]

Should be right, proved by 1950 probably. Charles Matthews 20:43, 10 February 2006 (UTC)[reply]

Projective linear group#Modular group[edit]

Title beautified. —Nils von Barth (nbarth) (talk) 04:02, 16 June 2010 (UTC)[reply]

The discussion in the subsection "modular group" here contains some errors. Some of the matrices listed as being elements of PSL(2,Z) are in fact only in PGL(2,Z). Tkuvho (talk) 16:07, 14 June 2010 (UTC) The error was introduced in an edit listed as "Revision as of 02:34, 2 November 2009". Tkuvho (talk) 16:13, 14 June 2010 (UTC)[reply]

If it helps, the edit was this one. The bottom row in the table all have determinant −1. JackSchmidt (talk) 05:48, 15 June 2010 (UTC)[reply]

Right. Torsion elements in PSL(2,R) correspond to elliptic transformations of the upperhalfplane, and would have to have fixed points in the interior of the upperhalfplane, whereas these have fixed points on the real axis. See related material at Cross-ratio#Six cross-ratios as Möbius transformations. Tkuvho (talk) 07:30, 15 June 2010 (UTC)[reply]

Oops, you’re completely right – thanks!

Somewhat subtly, the order 2 elements can be represented in PGL(2,Z) or in PSL(2,Z[i]) but not (as you note) in PSL(2,Z).

I’ve corrected this to correctly state which groups they are in, and give representative matrices – thanks for flag this to me.

—Nils von Barth (nbarth) (talk) 03:53, 16 June 2010 (UTC)[reply]

For reference, fixed in these edits (I also fixed formatting).

—Nils von Barth (nbarth) (talk) 03:56, 16 June 2010 (UTC)[reply]

One other note, for reference: the corresponding group in GL(2,Z) (order 12, covering the

S_{3}

in PGL(2,Z)) is the central extension (by −I; one may abusively call this the “covering

2\cdot S_{3}

”) – the integer matrices are a group in PGL, not in GL – since these just have 1, 0, and −1, this maps 1-to-1 into GL(2,Z/3), though not into GL(2,Z/2), where it’s 2-to-1 (b/c of 1 and −1). This group yields

C_{2}\times C_{3}

on intersection with SL(2,Z), then

C_{3}

on projection to PSL(2,Z), as you’d expect.

—Nils von Barth (nbarth) (talk) 04:26, 16 June 2010 (UTC)[reply]

Typos. —Nils von Barth (nbarth) (talk) 04:27, 16 June 2010 (UTC)[reply]

Action on p points ==[edit]

$(q^{n}-1)(q-1)=1+q+\cdots +q^{n-1}$ ???, this is wrong since q should be greater than 1 —Preceding unsigned comment added by 130.233.158.3 (talk • contribs) 2010-08-10T08:06:28Z

The missing division sign is now fixed. JackSchmidt (talk) 13:21, 10 August 2010 (UTC)[reply]

As an algebraic group[edit]

I think this claim:

PSL(n, K) and PGL(n, K) are algebraic groups of dimension n^2-1

is at best misleading and at worst wrong.

eg. http://mathoverflow.net/questions/16145/what-is-the-difference-between-psl-2-and-pgl-2/16175#16175 — Preceding unsigned comment added by 171.64.38.77 (talk) 23:40, 29 March 2012 (UTC)[reply]

Yes, this is definitely highly misleading. I have made a "temporary" change that acknowledges this issue. Hopefully it will be done properly at some point. I would cite the MathOverflow discussion, but I think that it doesn't count as a Wikipedia reference? There's also an explanation in Remark 9.3.4 of http://math.stanford.edu/~conrad/252Page/handouts/alggroups.pdf Quevenski (talk) 17:46, 3 December 2018 (UTC)[reply]

A typo in the section on exceptional isomorphisms?[edit]

One line claims that $\mathrm {PSL} (2,5)$ is isomorphic to $A_{5}$ , and the very next line contradicts this by claiming that $\mathrm {PSL} (2,5)$ is isomorphic to $S_{5}$ . We can make the second statement true by replacing $\mathrm {PSL}$ with $\mathrm {PGL}$ , but I do not know if it is an exceptional isomorphism.

Diagram with arrows[edit]

At the top of the article is a useful-looking diagram showing homomorphisms between various groups. The arrows indicating the homomorphisms vary: single heads and double heads, straight tails and curly tails. I do not know what these distinctions mean, and I do not know how to find out. I have searched Wikipedia and found no explanation, though I have found more such diagrams (e.g. at Schur multiplier).

I guess the best place for explanations of the arrow types would be at group homomorphism, but they're not there. Can anyone help? Maproom (talk) 08:29, 3 June 2013 (UTC)[reply]

Try Commutative_diagram#Symbols. Spectral sequence (talk) 20:22, 3 June 2013 (UTC)[reply]

The explanation for the arrows is the following: double heads mean that the homomorphism is surjective (onto), curly tails mean that the homomorphism is injective (one-to-one). Still this diagramme is misleading: In the upper right corner it looks like the homomorphism ends in a vector space, where it is meant that it ends in all elements that can be expressed as z^n. — Preceding unsigned comment added by 2A02:8070:8981:4800:D84D:F9D3:5565:D0DA (talk) 14:59, 13 February 2018 (UTC)[reply]

Projective_linear_group#Modular_group[edit]

This section claims that "the subgroup C3 < S3 consisting of the 3-cycles and the identity () (0 1 ∞) (0 ∞ 1) stabilizes the golden ratio and inverse golden ratio". I think this is wrong, the two stabilized values are $e^{\pm \pi i/3}$ , as the article for the Cross-ratio correctly claims. I'd edit it but I don't have the skills to edit the accompanying image. 84.121.59.163 (talk) 10:08, 3 February 2017 (UTC)[reply]

Unexplained notation[edit]

In the subsection Exceptional isomorphisms of the section Finite fields, suddenly the notation L_n(p^k) is used extensively without explanation. The explanation appears only at the beginning of the section Finite fields. This is too difficult to find if someone is just reading that subsection.

If it is necessary to use different notations for the same group in the same article, then please at least explain the new notation just before it is used. At the beginning of the subsection Exceptional isomorphisms would be a good place to explain it, or at least remind the reader of it.50.205.142.50 (talk) 19:34, 6 May 2020 (UTC)[reply]

Center[edit]

Using the notation Z(V) [resp. SZ(V)] for the homotheties in G = GL(V) [resp. G = SL(V)] and using the word "center" is misleading. It is true that the subgroup of homotheties contained in these groups coincides with their center, but 1. It is not true for any subgroup G of GL(V) (take G = diagonal matrices) 2. It has nothing to do with anything. The reason we quotient by the subgroup of homotheties is because they act trivially projectively (and not because they commute with everyone in G, who cares!). I think a terminology that would make sense is to call, for any subgroup G of GL(V), the projective group of G the quotient P(G) = G/(G \cap F^* id). This is the largest quotient of G that acts faithfully projectively. Note that this consists in taking the quotient of G by the equivalence relation "being nonzero scalar multiples", just like when defining a projective space. Taking the quotient of a group by its center is a different operation.