Talk:Symmetric group

message: Substituted at 02:37, 5 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
thread_title:  Modular group 

message: 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  Thanks, Jacob.Koot (talk) 15:49, 28 September 2018 (UTC)
replies: text: 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 Sn.  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)
replies: text:  I think mentioning the theorem with a reference would do. Jacob.Koot (talk) 10:28, 29 September 2018 (UTC)
replies: text:  20 months later, I finally got around to it. --JBL (talk) 16:54, 28 May 2020 (UTC)
thread_title:  minimal base 

message: "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)
replies: text: 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)
thread_title:  Not understandable 

message: 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)
replies: text: 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)
thread_title:  Illogical placement of definition of alternating subgroup?