# Talk:Alternating group

## 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)

## Notation

On a slightly more mundane note, this article uses both An and An. 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)

Since no objections have been raised in the intervening 28 months, go for it. --Vaughan Pratt (talk) 05:40, 12 October 2009 (UTC)
Done! (The roman/upright notation is presumably to distinguish from the group of Lie type, as at List of finite simple groups, but there is no ambiguity here.)
—Nils von Barth (nbarth) (talk) 03:13, 18 November 2009 (UTC)

## 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)

## 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)

The associated extensions $\operatorname {SL} _{n}(q)\to \operatorname {PSL} _{n}(q)$ are universal perfect central extensions for $A_{4},A_{5},A_{8}$ , by uniqueness of the universal perfect central extension; for $\operatorname {PSL} _{2}(9)\cong A_{6}$ , 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)

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

The bit about permutations in $A_{n}$ being conjugate in $S_{n}$ but not in $A_{n}$ is confusing to me. Is it because, in the examples that are given, every permutation by which permutation $p$ must be conjugated in order to obtain permutation $q$ is odd? If so, shouldn't the article mention this explicitly? 134.58.42.46 (talk) 13:37, 3 January 2012 (UTC)