Talk:Special linear group

The determinant of what now?

It says at the beginning that SL(n) is matrices with determinant one, but then that is the kernel of the determinant map. But that's wrong; it's the kernel of this map:

$A\mapsto \det(A)-1$

Am I wrong?

--Durka42 (talk) 01:22, 28 June 2013 (UTC)[reply]

Yes, SL(n) is the kernel of the determinant map. This is a map of groups, not a map of rings; the kernel is the preimage of the target group's identity. Sniffnoy (talk) 18:04, 28 June 2013 (UTC)[reply]

Thanks, I missed the multiplicative part. --Durka42 (talk) 03:48, 30 June 2013 (UTC)[reply]

Semidirect product

There is currently a section which states

GL(n, F) = SL(n, F) ⋊ F

Does anyone have a proof for this? I'm not convinced it is true. If F=ℝ and n is even, it seems there are elements of GL(n,ℝ) which can not be represented as a product of an element in ℝ and an element of SL(n,ℝ). Namely A ∈ GL(n,ℝ) with det(A) < 0. Perhaps I missed something. widdma (talk) 06:04, 29 August 2013 (UTC)[reply]

It's true, just not with the diagonal embedding (putting your field element on each spot in the diagonal); instead put your field element in the upper left corner and 1s on the rest of the diagonal. I agree that this should probably be made clearer, though. Right now it's stated only indirectly -- that embedding is what's meant by GL(1,F)→GL(n,F). Sniffnoy (talk) 20:19, 29 August 2013 (UTC)[reply]

Ah! I see now. Thank you. At the time I was also wondering why it was semidirect, as opposed to direct, because the diagonal embedding is normal. But your embedding is not normal, so all is well. widdma (talk) 23:58, 29 August 2013 (UTC)[reply]

Transvections don't always generate the special linear group?

The text in the section Special_linear_group#Relations_to_other_subgroups_of_GL.28n.2CA.29 explicitly says (but unsourced and without examples) that transvections do no always generate the full special linear group. However, I cannot see how this could fail to be the case. By Whitehead's lemma (linked from the section) diagonal matrices of determinant 1 do lie in the subgroup generated by transvections, and so do matrices that (upon left-multiplication) swap two rows while flipping the sign of one of them (so as to ensure determinant 1): they are easily a product of 3 transvections. It would then seem straightforward to apply Gauss elimination by row operations to a matrix in the special linear group, with a slight modification to ensure that any necessary permutation of scaling of rows is done in a "special linear" fashion. This allows bringing it into upper triangular form with entries 1 on the diagonal in all positions except possibly the final one, but that must be 1 too since we remain inside the special linear group. And the resulting upper unitriangular matrix is obviously in the subgroup generated by transvections. Could somebody confirm that the article needs correcting?

Marc van Leeuwen (talk) 12:45, 9 March 2014 (UTC)[reply]

Hm -- looking at the article, it does say "In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections"; I guess the preceding stuff was written assuming A was a general commutative ring? Which is at variance with the intro which assumes we're working over a field. So, it seems to me that's it's not so much wrong as it is unclear and badly organized. Sniffnoy (talk) 19:53, 9 March 2014 (UTC)[reply]

Another unclear point is the phrase "where SL(A) and E(A) are the stable groups of the special linear group and elementary matrices" in the end of that section; does it mean that elementary matrices are a group? Boris Tsirelson (talk) 06:49, 26 March 2015 (UTC)[reply]

I assume it means generated by them. But yeah that's unclear. Sniffnoy (talk) 18:43, 26 March 2015 (UTC)[reply]

Mistakes in "Relations to other subgroups of GL(n,A)"

> However, if A is a field with more than 2 elements, then E(2, A) = [GL(2, A), GL(2, A)], and if A is a field with more than 3 elements, E(2, A) = [SL(2, A), SL(2, A)].

This seems to be implying that $[GL(2,A),GL(2,A)]$ is not the same as $[SL(2,A),SL(2,A)]$ . But they are the same.