Talk:Symmetric group

Composition order

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 October 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)

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)

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)

Suggest earlier use of the term "permutation"

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)

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)

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)

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)

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)

Assessment comment

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.

For a topic as fundamental as the symmetric group, this is clearly in need of expansion. Silly rabbit 12:06, 13 May 2007 (UTC)

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

can we briefly deal with infinite case?

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)

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)

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)

@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)

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)

@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)

I would like to concur with Arided's overall point: infinite symmetric groups are important and should be discussed somewhere in wikipedia. I think the right place is in a section of this page, perhaps their own article someday - as it is now, its just the line in the intro saying they exist but won't be treated here as well as some facts about them sprinkled throughout the article. Let me note that the very first result on google for "infinite symmetric group" is a wikipedia page with the title "Infinite Symmetric Group," which really just links to this article, which is really about the finite case. While I agree that the notation Sym(Ω) need not appear in the lead, I do think the interested reader should be able to find their way to a section (a page?) with some basic information on infinite symmetric groups. I hope I will find the time to begin that section. Krb19 (talk) 15:12, 10 June 2021 (UTC)

Modular group

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

• ${\sigma _{i}}^{2}=1$
• $(\sigma _{i}\sigma _{i+1})^{3}=1$

and the presentation of the modular group:

• $S^{2}=1$
• $(ST)^{3}=1$

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)

minimal base

Every symmetric group Sm with m>3 has a minimal base of two elements. (finite m of course) If there exists literature or an article that contains this theorem, I would like to include the proof. Is there such literature or article? Jacob.Koot (talk) 11:19, 27 September 2018 (UTC)

There is a conventional two-element generating set for S_m mentioned in the article (section Generators and relations): any m-cycle together with a transposition that swaps two adjacent elements of the cycle. It would be nice to have a reference for this fact, as it is uncited in the article; I imagine it could be found in many texts on abstract algebra or combinatorics. --JBL (talk) 12:00, 27 September 2018 (UTC)

I can't find it in "Group Theory and its Applications to Physical Problems" of Morton Hamermesh. I have no other material at hand. Jacob.Koot (talk) 13:20, 27 September 2018 (UTC)

Browsing my bookshelf: Exercise 2.200 in Algebra by Mark Sepanski asks to show that (1, 2) and (1, 2, ..., n) generate S_n. The same statement is exercise 80δ in Elements of Abstract Algebra by Allan Clark, and is Exercise 6.6.16 in Algebra by Michael Artin. I did not find the statement in Algebra by Serge Lang. Introduction to The Theory of Finite Groups by Walter Lederman contains as exercise 3.8 the false statement that S_n is generated by (1, 2, ..., n) together with any transposition (one can easily check that this fails in S_4 when the transposition is (13)). So that's a bunch of good references for the fact, but sadly none that contain a proof. Undoubtedly a proof is written down in some reliable source somewhere. I will add one of these as a reference to the statement in the article in the near-but-not-immediate future. --JBL (talk) 14:58, 28 September 2018 (UTC)

Thanks, Jacob.Koot (talk) 15:49, 28 September 2018 (UTC)

Fraleigh has an exercise to prove this for S₅. I have done this exercise and have the proof in my notes. It would probably be easy to extend to S_n. The proof is rather long and I don’t have the time at the moment to put it into the article, but I could in a week or so. Do you think I should insert it?—Anita5192 (talk) 16:40, 28 September 2018 (UTC)

I think mentioning the theorem with a reference would do. Jacob.Koot (talk) 10:28, 29 September 2018 (UTC)

20 months later, I finally got around to it. --JBL (talk) 16:54, 28 May 2020 (UTC)

References

Not understandable

"Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives" The two matrizes above that line do not match that line. This needs a much better explained example. 2003:E7:2F09:BA28:AC82:73B0:DEBB:A4B8 (talk) 16:08, 22 July 2019 (UTC)

They do, in fact, match the sentence. I am willing to believe this could be more clearly written, but more details about what you find unclear would be helpful. --JBL (talk) 17:06, 22 July 2019 (UTC)

Illogical placement of definition of alternating subgroup?

In the section on normal subgroups, the finite case is discussed before the infinite case. In the finite case, no definition of an Alternating group is given while in the part on infinite groups, an explanation is given. In both cases, the alternating subgroup itself is finite, though a difference is that in the finite section it's called a group and in the infinite section, a subgroup (though in both cases its significance is as a subgroup). Wouldn't it make more sense to either do away with the definition in this section entirely or provide the definition in the first instance of the term in the part on normal subgroups? Correctorator123 (talk) 10:10, 13 January 2021 (UTC)

The definition of the alternating group in the finite case is in the subsection Symmetric_group#Transpositions,_sign,_and_the_alternating_group. (Previously, this section was just titled "Transpositions", making it hard to find the relevant information.) It is also discussed in the section Symmetric_group#Relation_with_alternating_group. So your summary of the situation does not seem particularly accurate. Separately, your assertion that the alternating group in the infinite case is finite is mistaken.
As far as navigation is concerned, I think the section heading "Elements" is very poor; does anyone have a better suggestion, given what it actually covers? --JBL (talk) 14:10, 13 January 2021 (UTC)