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Almost all modern published research in permutation groups uses fg to mean "apply f then apply g". I propose that the article use that convention and only note the other one as an exception. --Zero 03:57, 20 Oct 2004 (UTC)

I would tend to agree. I've come from reading the Schaum's Outline on Group Theory, and it was very disorienting to encounter the convention used here. --Paul 04:07, August 9, 2005 (UTC)

This text doesn't seem to follow: "The permutation f shown above is a cycle, since f(1) = 4, f(4) = 3 and f(3) = 1." What 'f' is it referring to? The only 'f' I see does not produce the results claimed here. Am I missing something obvious? --Paul 04:55, August 9, 2005 (UTC)

It is a vestige of earlier edits. Why don't you start on a clean-up (including the fg versus gf issue)? I'll back you up. --Zero 11:37, 9 August 2005 (UTC)[reply]

It's many months later, and I've just noticed your reply. Sorry for the delay, I got distracted. I may just do what you suggest. I need to get my bearings again.--Paul 05:04, 25 November 2005 (UTC)[reply]

Is it really true that fg means "apply f then g"? Unless I am confused I have always read it from right to left as in compositions of functions: "(f.g)(x) = f(g(x))". Also https://math.stackexchange.com/a/31764 seems to agree: "... so when you write "ab", you mean that you perform the permutation b first, and the permutation a second." 2001:16B8:40CA:3600:F1DE:E91:CA17:F2E6 (talk) 19:34, 29 January 2018 (UTC)[reply]

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)[reply]

Shouldn't the symmetric group be a functor Oset -> Grp rather than Set -> Grp? lars

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 for some n.

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

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

From the article as of 2011-06-29:

... the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions) from the set to itself...[1]

In the introductory paragraph, it would be easier for beginners (an audience wikipedia must serve) if the article used the relatively familiar word "permutation" before such unfamiliar phrases as "bijection from the set to itself" or "one-to-one and onto".

Note that the word "permutation" is used later in the text, under heading Elements:

The elements of the symmetric group on a set X are the permutations of X.

Because I am not intimate with group theory terminology and jargon, I am reluctant to perform such an edit unilaterally, for fear of misusing the idiom, but I would be glad to propose specific text for consideration, if anyone is interested. If no one responds, after several weeks I will do the edit myself regardless. Dratman (talk) 12:34, 29 June 2011 (UTC)[reply]

The problem is that the word permutation, while more familiar, is also more ambiguous, as you can check in that article: it often means more simply an ordering of the elements into a list rather than a bijection of the set to itself. It makes fairly little difference if the set is {1,2,...,n}, but a big difference in other cases. In order to define composition, which is central here, one needs bijections. So a reader who would wonder how to compose permutations of an arbitrary set would have to look up permutation, and find the term "bijection" on his path there anyway. We might as well be frank about this from the start. Marc van Leeuwen (talk) 04:39, 30 June 2011 (UTC)[reply]

I made an edit similar to the one I proposed below, trying to balance the usefulness of the well-known idea of a permutation with the requirement for precision. But more recently, in playing around with examples of the symmetric group, I am even more struck by your point about ambiguity in the concept of a permutation. Clearly "bijection" is the right tool to use -- yet even "bijection" is not without problems. The domain and codomain of a bijection, as of any function, must consist of named or otherwise identifiable and distinguishable elements. Otherwise we can't specify or describe the function in any way. In (most?) other areas, domains and codomains are fixed rather than indefinite. But such mutability when compared with more ordinary references to functions, I think, makes notation for the symmetric group more confusing, regardless of whether we write about bijections or permutations. Dratman (talk) 01:09, 3 August 2011 (UTC)[reply]

I understand and, in general terms, concur with your objection to "permutation." The word "bijection" is more precise. That said, I still wish to argue that it would be better to use the more familiar term in the first paragraph of the article. (Since we are discussing pedagogy rather than mathematics, there is no single provably right answer here.)Dratman (talk) 01:09, 3 August 2011 (UTC)[reply]

Before suggesting any replacement text, I want to quote from the first sentences of the current definition at Mathworld:

The symmetric group Sn of degree n is the group of all permutations on n symbols. Sn is therefore a permutation group of order n!

Notice that the introductory Mathworld sentence omits any mention of "on a set." That is correct, because the symmetric group is not related to any particular "set X" consisting of n elements. Only the cardinality of the set of symbols matters.

I propose the following rough draft replacement for the first sentence of the article:

In mathematics, the finite symmetric group Sn, for n a positive integer, is the group consisting of all possible permutations (rearrangements) which can be carried out on a set of size n. The order (number of elements) of the symmetric group is therefore n! The group operation consists of carrying out two successive permutations, whose combined effect is necessarily equivalent to that of a single permutation which is (as required by closure under the group operation) also an element of the group. Dratman (talk) 03:26, 1 July 2011 (UTC)[reply]

The comment(s) below were originally left at Talk:Symmetric group/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Last edited at 12:06, 13 May 2007 (UTC). Substituted at 02:37, 5 May 2016 (UTC)

Since infinite symmetric groups can be defined, I think that should be explained in this article, in order to be fully "encyclopedic". I added a "Why?" tag in the lede at the appropriate point to signal this view. The general definition is just this: "The symmetric group Sym(Ω) on a set Ω consists of all bijections from Ω to Ω under composition of functions." That might as well be included in the article; then, infinite and finite cases could be briefly contrasted. Arided (talk) 14:40, 29 January 2018 (UTC)[reply]

I changed the 1st sentence of the lede accordingly, but haven't built in the short compare-and-contrast paragraph that would resolve the "Why?" Arided (talk) 14:44, 29 January 2018 (UTC)[reply]

I would much prefer that you return the lead to the way it previously was, when it accurately summarized the article. As it is, the lead begins with a piece of notation not otherwise used in the article, and suffers from other infelicities (the redundant invocation of bijections, the improper use of Template:why).

Instead, I invite you to propose (either on this talk page, or by adding it directly to the article) a section called "Infinite symmetric groups". Once such a section exists, we can adjust the lead to properly summarize the article contents. --JBL (talk) 18:18, 29 January 2018 (UTC)[reply]

@JBL, notation isn't the central part of my suggestion. The suggested sentence can be easily rephrased without any special notation:

The symmetric group defined over any set consists of all bijections from the set to itself under composition of functions.

Or similar. Since that's the definition of "symmetric group", it seems suitable for the first sentence of the article. I don't think it absolutely needs a new section. (Presumably it would be possible to write a section on infinite symmetric groups, I'm not the best person to do that.) I suggest we revise the lede to match the quote above, then leave the new section you mentioned to some future contributor. Arided (talk) 18:33, 12 February 2018 (UTC)[reply]

Also, the discussion above at Talk:Symmetric_group#Suggest_earlier_use_of_the_term_"permutation" is relevant -- we seem to be rehashing that discussion here. My basic point is that since this is a mathematics article it should actually include the definition of the term it's about, prominently. Finite groups are a special case of that definition. Arided (talk) 18:41, 12 February 2018 (UTC)[reply]

@Arided: I have made some changes relating to your last edit, moving words and links around between the first two sentences. I hope you find the result in keeping with your intent. I have removed the template:why again: the template is for statements that require clarification. The sentence in question is perfectly clear; what you really intend is not "what does this mean?" but rather "this seems like an unfortunate state of the universe," and I don't know if there's an appropriate tag for that. --JBL (talk) 21:51, 16 February 2018 (UTC)[reply]

There's an "obvious resemblance" between the generators (from the middle of this article):

and the presentation of the modular group:

There's some fun and games one can accomplish due to this resemblance, but I no longer recall the details. I'm probably mis-remembering something, but there's an analogy to the way in which sl(2,C) is the prototypical semi-simple Lie algebra, with raising and lowering operators that can be combined together to make the root diagrams of other Lie algebras, and, when combining an infinite number of these, to get the affine Lie algebras. Something to do with braid groups, to be more specific. I can't recall/reconstruct in my head what the analogy is; could more info be posted on this? 67.198.37.16 (talk) 00:05, 8 March 2018 (UTC)[reply]