Talk:Permutation group

message: I find the (very brief) introduction of cyclic form still rather opaque.  We ought to aim for a clearer explanation of what, for example,  " (1,2,4)(3)"  means.  
152.3.68.83 (talk) 14:33, 25 July 2013 (UTC)
thread_title:  Article needs more exposition 

message: I added two examples from my own work.  Others may argue that this is too egocentric. Cullinane 21:54, 10 August 2005 (UTC)
thread_title:  Examples of permutation groups 

message: I wonder if a more exact definition of a permutation group would be something along the lines of "A permutation group is a pair (G, M) where M is a set and G is a group acting on M; when no ambiguity can arise, it may be simply denoted by G.  Two permutation groups are said to be isomorphic (as permutation groups) provided there exists a map blah blah blah such that blah blah blah".
Or, perhaps "A permutation group is a group G together with a set M such that G acts on (or permutes) M."
See the way topological space is defined in Topological spaces, which somehow seems both more natural and more exact.
(Of course, if every group theory book in the world does it the way the article does, then there's no leeway to make a change.)
Son of eugene (talk) 08:51, 31 December 2010 (UTC)
thread_title:  Definition 

message: Perhaps this is a rather silly question for those who are familiar with the subject, but what is the result when a permutation g =(1 2 3 4)|g \in G acts on A={1,2}? This operation should be possible: we can construct G from {1,2,3,4}, abstract it out into a group, and make it act on {1,2}. I am confused because what (1 2 3 4) is telling me is that move the 2nd element to the 3rd position, but there is no 3rd position! In particular, If i assume that gA={1}(i.e that 2 is chucked out of the set), then what about g3=(1 4 3 2) acting on A? What I mean is, {}=g2(gA) ≠ (g3)A={2}, which violates the associativity of group action! Pratik.Mallya Talk! 23:32, 2 March 2011 (UTC)
replies: text: I think perhaps the concrete example of permutation, as I have it here, can only apply to the set from which they were originally constructed? Pratik.Mallya Talk! 23:43, 2 March 2011 (UTC)
replies: text:  In order for a group G to act on a set S, each element g of G must map S to S. In other words, the set S must be fixed setwise by action of any group element. So, in your case, G does not act on A, since the permutation g does not send elements of A to elements of A. See group action. -Krasnoludek (talk) 00:05, 24 March 2011 (UTC)
thread_title:  (1 2 3 4) on {1, 2} 

message: I feel it would be worthwhile to mention Cayley's Theorem  (dedicate a section/ just link) at some point on this page. To me this is the reason that these groups are so important...
--ShmulikG (talk) 15:42, 11 March 2013 (UTC)
thread_title:  Cayley's Theorem 

message: I have temporarily moved a recent edit here. The moved section is:
While we can think of a single permutation as a rearrangement of our objects, 1,2,3 and 4 in the present case, it is easier to understand permutation groups if we think of a permutation P of n objects as a mapping of the set of objects onto itself. We can present it as a table
:P = \tbinom{a_1\ a_2 \ \dots \ a_n}{b_1\ b_2\ \dots \ b_n}.
If Q  is another permutation
:Q = \tbinom{b_1\ b_2\ \dots \ b_n}{c_1\ c_2\ \dots \ c_n}\ 
the composition or product of the permutations is defined as
:\ PQ = \tbinom{a_1\ a_2\ \dots \ a_n}{c_1\ c_2 \ \dots \ c_n}.
Note that the convention for the composition of permutations is the opposite of the convention used when forming the composition f∘g of two functions. In PQ we apply P first and Q afterwards, but when we form f∘g we apply the second function g first.
Example. Let
:P = \tbinom{1\ 2\ 3\ 4}{2\ 1\ 4\ 3},\ \ Q = \tbinom{2\ 1\ 4\ 3}{2\ 3\ 4\ 1} = \tbinom{1\ 2\ 3\ 4}{3\ 2\ 1\ 4}.
Then PQ = \tbinom{1\ 2\ 3\ 4}{2\ 3\ 4\ 1}  and  QP = \tbinom{1\ 2\ 3\ 4}{4\ 1\ 2\ 3}.
PQ is also called the product of the group elements P and Q. As we just saw, this kind of multiplication is not usually commutative.
I think that this attempted rewrite is meant to be a gentler explanation of what it replaces and I am in favor of that, but it is not up to the standards for WP articles. I thought that it would be better to point out the problems and fix them here before putting it back into the article.
* One problem is the non-encyclopedic tone of the passage created by the constant use of the pronoun "we". 
* The first sentence refers to a four element set, but the rest of the paragraph talks about a general n element set. 
* The sentence about the order of composition is misleading. Both conventions are in common use and the one being touted here (while my personal preference) has been losing ground for the past few decades. 
* There are no citations given in the section and while they are not needed here for verifiability, they are needed to give readers a place to go to get more information. 
These problems need to be fixed. A less serious issue is the use of the two-line notation for permutations. While not incorrect, any reference on permutation groups is going to use cyclic notation and a reader, starting from this page, will not be prepared for that. I suggest using both notation styles initially and then phase out the two-line notation. I also think that the font size used by the \tbinom command is a bit small for this display (especially for the general expessions). Bill Cherowitzo  (talk) 05:25, 3 May 2014 (UTC)
thread_title:  Examples section 

message: The Neutral element and inverses section contained the statement:
:"In cycle notation one can reverse the order of the elements in each cycle to obtain a cycle notation for its inverse. Thus,
:: [(1 2 5)(3 4)]^{-1} = (5 2 1)(4 3) = (1 5 2)(3 4)."
This is a case of the wrong procedure leading to the correct result. It's correct in this case because (125) and (34) have no common element, and so they commute. This is not true in general for permutations with common elements. For example:
: [(12)(13)]^{-1} = (13)(12) \ne (12)(13)
The error here may have been to assume that cycle notation only applies to the orbits of a permutation, but it doesn't. (125)(34) can also be written as (15)(12)(34) -- a product of 2-cycles. This is also in "cycle form", but its first two terms don't commute. --Stfg (talk) 12:53, 23 July 2015 (UTC)
thread_title:  Howler 

message: Substituted at 02:28, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Cayley's theorem says there is a 1 to 1 correspondence between groups and subgroups of Sn=permutation groups.
As we know, permutation groups can be thought as subgroups of Sn. That means, by Cayley theorem, they are in 1 to
1 correspondence with axiomatic groups. This is extremely important to notice, because it connects this not-so used today notion directly to a fundamental concept taught today in math degrees and courses.  — Preceding unsigned comment added by Santropedro (talk • contribs) 20:23, 17 January 2017 (UTC)
thread_title:  Complete Cayley theorem 

message: I'm proposing to merge Rank 3 permutation group, Oligomorphic group, and Primitive permutation group into this article. Each of the 3 articles is quite short, with Rank 3 permutation group consisting largely of examples and both Oligomorphic group and primitive permutation group being stub class articles. Each of these three articles are specific cases of permutation groups, and would make the most sense in the context of permutation groups at large. --TripleShortOfACycle (talk) 05:28, 17 May 2020 (UTC)
replies: text: Oligomorphic group is a teensy article and merging seems like a no-brainer.  The material at Primitive permutation group also seems like it could be comfortably covered here.  The table at Rank 3 permutation group is really enormous, and I do not think that at present that article fits so well with this one -- there is no discussion of transitivity or of rank in this article.  (I would be inclined to call that a significant oversight or failing of the present article that should eventually be fixed; but as long as it's not fixed, the content of Rank 3 permutation group fits poorly.) Because of the huge table, I would suggest at most a short gloss with a  link. --JBL (talk) 12:52, 29 May 2020 (UTC)
replies: text:  I have merged Oligomorphic group here.  --JBL (talk) 14:38, 10 June 2020 (UTC)

text:  I changed my mind about whether Primitive permutation group can be included here: it's a short article, but it's heavily technical.  Instead, I have included a short definition and example here, with a pointer back to the main article.  I will try to do the same thing with rank eventually. --JBL (talk) 14:36, 5 July 2020 (UTC)
replies: text:  I think that's probably the right thing to do. I've been contemplating the merge for the past few days and I didn't like the amount of work needed to reduce the technicality so that it would fit in this article. I also agree that the rank 3 article should remain separate, that table would never fit this article. --Bill Cherowitzo  (talk) 19:41, 5 July 2020 (UTC)
thread_title:  Merger Proposal 

message: https://en.wikipedia.org/wiki/Symmetric_group
I believe they talk about the same topic? 100.36.227.237 (talk) 02:36, 7 October 2021 (UTC)
replies: text: No: Read the first paragraph of this article. The symmetric group is a specific permutation group. It has many important properties that are not shared by other permutation groups, and justify a specific article. D.Lazard (talk) 10:00, 7 October 2021 (UTC)

text: No, certainly not. --JBL (talk) 10:09, 7 October 2021 (UTC)
thread_title:  Merger with Symmetric group 

message: The article says (and some other articles are similar):
: There are two ways to denote the composition of two permutations. In the most common notation, \sigma\cdot \tau is the function that maps any element x to \sigma(\tau(x)). The rightmost permutation is applied to the argument first [cite to Biggs]A different rule for multiplying permutations comes from writing the argument to the left of the function, so that the leftmost permutation acts first.[cites to Dixon&Mortimer, Cameron and Jerrum] ...In this notation, the permutation is often written as an exponent, so σ acting on x is written xσ; then the product is defined by x^{\sigma\cdot\tau} = (x^\sigma)^\tau. This article uses the first definition, where the rightmost permutation is applied first.
I work with permutation groups quite often and I'm pretty sure I have never seen the notation \sigma\cdot \tau. In particular, the source Biggs doesn't use a dot, but just writes \sigma\tau. None of the three cites for the second rule use a dot either. So the dot has to go.I also object to "most common notation" as it is OR and unprovable. Maybe in some fields of mathematics or science it is the most common, but within the domain of group theory it is probably the least common. I did a quick survey of books: Here is the image of x under "\sigma\tau" :
right-to-left inline \sigma\tau x=(\sigma\tau)(x)=\sigma(\tau(x)) Kerber, Representations of Permutation Groups (1971); Biggs&White, Permutation Groups and Combinatorial Structures (1979).
left-to-right inline x\sigma\tau = (x)(\sigma\tau)=((x)\sigma)\tau Passman, Permutation Groups (1968); Cameron, Permutation Groups (1999).
left-to-right exponential x^{\sigma\tau} = (x^\sigma)^\tau Wielandt, Permutation Groups (1964); Dixon&Mortimer, Permutation Groups (1996); Bhattacharjee et al, Notes on Infinite Permutation Groups (1998); Holt et al, Handbook of Computational Group Theory (2005); Huppert, Endliche Gruppen I (1967); Praeger&Schneider, Permutation Groups and Cartesian Decompositions (2018); Seress, Permutation Group Algorithms (2003); Encyclopedia of Mathematics (2002+).
A permutation group specialist that I discussed this with believes that the preference for the exponential notation in that field derives from the enormous influence of Wielandt's little book (and also that it works very well). As I said, preferences may be different outside that field. I plan to reword that section of the article to lead with the exponential notation and mention the other two as alternatives. I'll wait a short while for comments first. McKay (talk) 06:08, 12 April 2024 (UTC)
thread_title:  Notation for composition