Talk:Alternating group

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[edit] Isomorphism of PSL(2,4), PSL(2,5), and A5

I have trouble believing that PSL(2,F_4) is isomorphic to PSL(2,F_5), because I think the size of PSL(2,F_q) depends in a simple way on q that rules this out. But I'm very jetlagged right now so maybe I'm mixed up! Can anyone clear this up?

John Baez 8 July 2005 00:29 (UTC)

PSL(2,F_4) has order (4²-1)*4=60 and permutes 5 rays. Thus it is isomorphic to A₅.

PGL(2,F_5) has order (5²-1)*5=120 and permutes 6 rays. PSL(2,F_5) is the subgroup in PGL(2,F_5) of even permutations, order 60. It permutes 5 of its cosets in A₆, is thus isomorphic to A₅. Scott Tillinghast, Houston TX (talk) 05:08, 26 December 2007 (UTC)

Okay, thanks for clearing that up. I tend to forget (and then remember, and then forget) the sizes of GL(n,F), PGL(n,F), SL(n,F), and PSL(n,F), and how they depend on whether or not F has a square root of -1. John Baez (talk) 19:06, 4 February 2008 (UTC)

[edit] Notation

On a slightly more mundane note, this article uses both A_n and A_n. I have never seen the former; Dummit and Foote's Abstract Algebra, Michael Artin's Algebra, and Lang's Algebra all have the latter. I shall make the appropriate changes unless there are any objections. Xantharius 18:22, 8 June 2007 (UTC)

[edit] Schur multipliers

Schur multipliers of alternating groups are calculated in Aschbacher's text on Finite Groups. In particular it contains the following two lemmas: The exponent of the Schur multiplier divides the order of the finite group. If a Sylow subgroup of the group is cyclic, then the corresponding Sylow of the Schur multiplier is trivial. Robinson's text on group theory contains the following lemma: The Sylow subgroup of the Schur multiplier of a group is a quotient of the Schur multiplier of the Sylow subgroup.

Since SL(2,3) is a stem extension of PSL(2,3)=A4, and the Schur multiplier of C2 x C2 is C2, the Schur multiplier of A4 is C2. One could also just work out the homology, the cohomology, and the Hopf formula by hand for A1=A2=1, A3=C3, and A4. JackSchmidt 02:44, 17 October 2007 (UTC)

[edit] Extension of A₈?

The group of scalar matrices in GL(4,2) is trivial, so this is not a way to construct a central extension of A₈. Scott Tillinghast, Houston TX (talk) 05:06, 1 January 2008 (UTC)

This is about the paragraph

The associated extensions are universal perfect central extensions for A₄,A₅,A₈, by uniqueness of the universal perfect central extension; for , the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group.

It may be wise to place in a new section since there is another error making the correct version somewhat irrelevant. A₄ is not perfect, so it has no universal perfect extension. As you noted, A₈ has a twofold perfect cover, but it is not SL(4,2)=PSL(4,2)=A8. Until someone salvages the good content, I commented out the text on the article page. JackSchmidt (talk) 06:53, 1 January 2008 (UTC)

Here is a construction on A₆:

PSL(3,4) has a subgroup isomorphic to A₆. Denote its pre-image in SL(3,4) as 3.A₆. This is not a direct product, because its 3-Sylow subgroups are non-abelian, order 27.

Consider the direct product of 3.A₆ and SL(2,9). Take the subgroup consisting of pairs in which both elements correspond to the same element in A₆. Scott Tillinghast, Houston TX (talk) 21:11, 3 January 2008 (UTC)

I have checked out Karpinsky's book 'The Schur Multiplier' but have to return it Monday. I am making notes on his construction of central extensions of symmetric and alternating groups. I will see whether I can describe them clearly in a way that will fit into this article. Scott Tillinghast, Houston TX (talk) 04:51, 16 February 2008 (UTC)

[edit] A source for the conjugacy classes ?

I would add the data on the conjugacy classes to the French article. These data are easy to prove, but I don't find them in my textbooks and I would like to give a reference. Can anybody indicate a reference ? (A source in English is good, of course.) Thanks. Marvoir (talk) 10:32, 5 January 2008 (UTC)

One affordable reference is (Scott 1987, §11.1, p299).

• Scott, W.R. (1987), Group Theory, New York: Dover Publications, ISBN 978-0-486-65377-8 

Thank you very much ! Marvoir (talk) 09:59, 6 January 2008 (UTC)