Talk:Alternating group

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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 00:29, 8 July 2005 (UTC)[reply]

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)[reply]

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)[reply]

And what about the isomorphism of PSL(2,3) with PSL(7,2), the simple group of order 168 ?— Preceding unsigned comment added by 2601:204:f181:9410:ac2a:f9f1:3e4d:baf6 (talk) 18:25, 9 December 2024 (UTC)[reply]

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)[reply]

Since no objections have been raised in the intervening 28 months, go for it. --Vaughan Pratt (talk) 05:40, 12 October 2009 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

This is about the paragraph

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

dead references

The link in footnote 1 is now dead. It should probably be repointed at the current book available from Springer, since the author had pulled all the older copies. Alternatively, some other reference should be used. —Preceding unsigned comment added by 98.174.185.105 (talk) 15:55, 6 June 2010 (UTC)[reply]

Conjugacy classes

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)[reply]

External links modified

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I checked the archives, but the referenced work, "Chapter 2: Alternating groups" no longer exists in any of the archives. — Anita5192 (talk) 18:02, 27 February 2016 (UTC)[reply]

Alternating

Hi all, I would be happy if the article would explain the name 'Alternating' for this group. As long as this is unfortunately not the case, maybe somebody is willing to explain it on this talk page? Regards, Bob.v.R (talk) 20:33, 25 March 2016 (UTC)[reply]

I think this is a very good suggestion! Here: https://mathoverflow.net/a/74220/103598 is a mathoverflow post answering this question. Maybe someone could work this up to a small section on etymology on this wikipedia page? Zaunlen (talk) 15:23, 10 November 2019 (UTC)[reply]

Definition of simple group is ignored

In the section Basic properties, this sentence appears:

"The group A_n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5."

But the definition of "simple" at the linked article applies to any group G whose only normal subgroups are the trivial group and G itself.

This definition applies to the alternating groups A₁ and A₂, as well as A₃, and A_n for n ≥ 5.

In other words, for all n ≠ 4.

It seems inappropriate to use here a definition of a "simple group" that excludes abelian groups from being simple.

I hope someone familiar with group theory will fix this. 2601:204:F181:9410:5DBA:9094:DC0B:FD72 (talk) 15:35, 8 December 2024 (UTC)[reply]

The groups A₁ and A₂ are trivial, and the trivial group is not considered a simple group, as the first sentence of the Simple group article makes clear. The group A₃ is abelian, so the article is not excluding abelian simple groups. --Zundark (talk) 20:53, 8 December 2024 (UTC)[reply]

Thank you for kindly explaining where I was off base. I hereby retract any objection expressed above, and I apologize for my errors. 2601:204:F181:9410:AC2A:F9F1:3E4D:BAF6 (talk) 02:10, 10 December 2024 (UTC)[reply]