Talk:Symmetric group/Archive 1

Archive 1

More material and editing needed

Given the vast amount of material available on the symmetric group I find this article hardly adequate. It spends much time on explaining trivialities and little on explaining actual properties. There are books like Bruce Sagan's Symmetric Group dedicated to the subject. The representation theory is also very rich but available elsewhere without proper links. I added a couple of basic pieces but don't have a clear understanding if the community even cares about this. Mhym 21:34, 22 January 2006 (UTC)

permutation group

nobody seemed interpelled by my comment on Talk:Permutation, shouldn't et least permutation group and symmetric group better be merged? — MFH:Talk 19:33, 10 November 2006 (UTC)

Why should they be merged? A permutation group is in effect an injective homomorphism of any group to a symmetric group. But the group theory of the symmetric group is not particularly closely related to the basic discussion of orbits, and so on? Charles Matthews 17:59, 11 November 2006 (UTC)

Cycle Graphs

Something this whole website seems seriously lacking in is the cycle graphs of the symmetric groups. I can't seem to find them anywhere in fact. Could somebody draw some and put them on this page?--SurrealWarrior (talk) 03:18, 20 April 2008 (UTC)

None of the cycle graphs for S3, S4, or S5 have overlapping cycles, so they are determined simply by the number of cycles of each length. The cycle graph of S3 has three cycles of length 2 and one cycle of length 3. The cycle graph of S4 has six cycles of length 2, four cycles of length 3, and three cycles of length 4. S5 has fifteen cycles of length 4, six of length 5, and ten of length 6. S6 has over two hundred cycles and they overlap. JackSchmidt (talk) 03:45, 20 April 2008 (UTC)

Big monster and little group

I seem to remember two large finite symmetric groups called big monster and little monster — is that right? Are they relevant? m.e. 15:34, 11 November 2006 (UTC)

No, those aren't symmetric groups. Charles Matthews 17:56, 11 November 2006 (UTC)

Yes they are - every group is a subgroup of the symmetric group, and thus a symmetric group, by Cayley's Theorem. However, they aren't actually relevant per se. 82.9.63.89 (talk) 16:57, 7 April 2008 (UTC)

Subgroups of symmetric groups are not themselves necessarily symmetric groups. There is no set such that either of the "monster groups" is its symmetric group, so neither is a symmetric group. —Simetrical (talk • contribs) 01:15, 8 April 2008 (UTC)

I've seen this terminology misunderstanding before. A subgroup of a symmetric group is called a permutation group. A finite symmetric group is the group of all permutations of a finite set, so its order is always of the form n! for n the size of the set. A permutation group acts on a set of size n, but it may not contain every permutation of that set. A symmetry group is easily understood as a special sort of permutation group, and there are versions of Cayley's theorem that describe any group as the symmetry group of an object in some fixed class of things-with-symmetry-groups. One of the easiest versions of this is the representation of a group as the centralizer of its regular representation. The articles on the sporadic groups tend to indicate how these groups are thought of as symmetry groups. JackSchmidt (talk) 03:53, 20 April 2008 (UTC)

Ordered sets?

Shouldn't the symmetric group be a functor Oset -> Grp rather than Set -> Grp? lars — Preceding unsigned comment added by 80.128.122.101 (talk) 07:14, 4 June 2006 (UTC)

No – one takes the symmetric group of a set, without needing an order. For example, the symmetric group of {1,2} is the same as the symmetric group of {2,1} – they’re both {(),(12)}.

However, you’re right that orderings are very closely related – once one orders a set, then one can identify the symmetric group of that set with (other) orderings, and indeed one can identify the set with an ordinal and hence identify the symmetric group of the set with $S_{n}$ for some n.

—Nils von Barth (nbarth) (talk) 09:51, 24 November 2009 (UTC)

Incorrect statement

The statement (in the section Applictions):

"The symmetric group on a set of size n is the Galois group of the general polynomial of degree n" is incorrect! The Galois group of a polynomial of degree n is a subgroup of the symmetric group. It is unclear what does mean "the general polynomial of degree n". VictorMak (talk) 15:42, 9 March 2011 (UTC)

This is a bit subtle. The general polynomial of degree n over a field K is a polynomial that is actually defined over the field of fractions K(α₀,...α_n−1), and it is given by Xⁿ+α_n−1Xⁿ⁻¹+...+α₀. In other words it is a polynomial with indeterminate coefficients. It is this polynomial that always has as Galois group the full symmetric group. See Abel-Ruffini theorem and its talk page. Marc van Leeuwen (talk) 08:46, 10 March 2011 (UTC)