Talk:Mathieu group

message: The constructions of M12 and M24 that I have described may have been Mathieu's original constructions, but I have not had access to his papers. Scott Tillinghast, Houston TX 03:17, 7 June 2007 (UTC)
thread_title: Constructions

message: I am thinking about making some internal links in this article. Some of the maximal subgroups listed at the end are relevant in earlier sections Scott Tillinghast, Houston TX (talk) 05:01, 5 May 2008 (UTC)
thread_title: Internal links

message: This article is written at a level that only advanced mathematicians can understand.  At the very least, the article needs an introduction that is more easily understood by a general audience.  Dr. Submillimeter 10:41, 30 March 2007 (UTC)
thread_title: Language problem

message: It has been requested of me to summarize the sources of where I get the material, as an aid to others who want to edit.
===Automorphism groups of Steiner systems===
This comes mostly from Dixon and Mortimer's Permutation Groups. The development is bottom-up: start with the quaternion group and PSL(3,4) (also called M21) as nest eggs and work upward.
===Automorphism group of the Golay code===
From Thomas Thompson's Carus monograph (especially appendix) and Robert Griess's Twelve Sporadic Groups (chapter 5).
===Sextet subgroup===
This comes from Robert Griess's Twelve Sporadic Groups, chapter 4. His treatment is quite sophisticated, so I worked out my own presentation.
I found I could make a link to the hexacode.
I was considerably exposed to a socratic way of teaching, so I have left you, dear readers, with a set of generators and you can play around with the sextet group as you like.
For the points I used alphabetic characters. They have nothing to do with the alphanumeric characters I used in an earlier section to describe generating permutations.
===Maximal subgroups===
Robert Griess in chapter 6 describes the maximal subgroups of M24 and refers to the 1977 paper of Robert T. Curtis.
Conway and Sloane list on one page the maximal subgroups of all 5 Mathieu groups.
Then there are the ATLAS webites of Robert A. Wilson et al. They list all the maximal subgroups of Mathieu groups.
You may wonder why I have omitted maximal subgroups of M12 and M11. I have worked on text but I am in a quandary and want to think about it. I started on Mathieu groups with M11 and worked bottom-up, and liked doing it that way. The listing of maximal subgroups is top-down. So I am not sure what comments to make about subgroups of M11 and M12. M11 has permutation representations of degrees both 11 and 12!
Scott Tillinghast, Houston TX (talk) 04:56, 5 May 2008 (UTC)
thread_title:  Summary of sources, aids to editors 

message: Robert Griess in chapter 6 describes the maximal subgroups of M24 and refers to the 1977 paper of Robert T. Curtis.
Conway and Sloane list on one page the maximal subgroups of all 5 Mathieu groups.
Then there are the ATLAS webites of Robert A. Wilson et al. They list all the maximal subgroups of Mathieu groups.
You may wonder why I have omitted maximal subgroups of M12 and M11. I have worked on text but I am in a quandary and want to think about it. I started on Mathieu groups with M11 and worked bottom-up, and liked doing it that way. The listing of maximal subgroups is top-down. So I am not sure what comments to make about subgroups of M11 and M12. M11 has permutation representations of degrees both 11 and 12!
Scott Tillinghast, Houston TX (talk) 04:56, 5 May 2008 (UTC)
thread_title: Maximal subgroups

message: I didn't like this phrase: It follows from the classification of finite simple groups that the only groups which are k-transitive for k at least 4 are the symmetric and alternating groups (of degree k and k-2 respectively)  
This is easily misunderstood: A hurried reader may read this as an assertion that the alternating group Ak-2 is k-transitive.
Suggested rephrasing: It follows from the classification of finite simple groups that the only groups which are k-transitive for k at least 4 are the symmetric groups Sk and the alternating groups Ak+2 
Cool dude ragnar (talk) 06:57, 18 May 2008 (UTC)
replies: text: Sounds good. JackSchmidt (talk) 00:28, 19 May 2008 (UTC)
replies: text: Fixed (again?). JackSchmidt (talk) 02:44, 16 May 2010 (UTC)

text: I adjusted this in a few spots.  There might still be more. JackSchmidt (talk) 02:44, 16 May 2010 (UTC)
thread_title: Multiply transitive phrasing

message: I suggest putting conjugacy classes that are not power equivalent on separate lines. Classes may not be of equal size. Another thing to list would be permutation structures of the conjugacy classes. Perhaps room for that could be made by eliminating the factorizations of the orders of elements.
Listing the conjugacy classes suggests listing character tables, but that could make for a pretty long article, and M24 has a 26-by-26 character table. Scott Tillinghast, Houston TX (talk) 21:51, 30 June 2008 (UTC)
replies: text: John McKay  using a fuller notation for the cycle structure in the "Cycle structure and conjugacy" column of the M24 table in the element section. For instance, instead of 5^4 use 5^4 1^4 to explicitly indicate the fixed points. JackSchmidt (talk) 17:03, 22 September 2009 (UTC)
thread_title:  Tables of elements of given order 

message: The first paragraph says "five finite simple groups ... M11, M12, M22, M23, M24."  But the first table in the article lists M21 as simple.  Later we are told that M21 is a name for PSL(3,4), which also implies that it is simple. The list at List_of_finite_simple_groups omits M21, but that would be correct, as M21 is not sporadic, it is a linear group. I would change the first paragraph, if I were more confident, to read either " ... six finite simple groups ... M21 ..." or "... five sporadic finite simple groups ...".  Maproom (talk) 10:28, 23 July 2010 (UTC)
replies: text: See the second paragraph of the intro. JackSchmidt (talk) 12:41, 23 July 2010 (UTC)
replies: text:  Ok - I won't argue.
Another point: M7 is the trivial group, and is therefore simple. But the table lists it as "not simple".  Is this worth correcting? Maproom (talk) 17:40, 26 July 2010 (UTC)

text: The trivial group isn't usually considered simple, for much the same reason that 1 isn't usually considered a prime number. --Zundark (talk) 13:20, 23 February 2011 (UTC)
thread_title:  Is M21 a Mathieu group? 

message: Could anybody give a reference for the following statement, please ? "The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at least 6, and the Mathieu groups M24, M23, M12 and M11. The full proof requires the classification of finite simple groups, but some special cases have been known for much longer."
I don't deny it, but I would like to give a reference for the corresponding statement in the French article. Thanks. Marvoir (talk) 12:05, 3 March 2012 (UTC)
replies: text: I've added a reference. Maproom (talk) 10:49, 4 March 2012 (UTC)
thread_title:  Request for a reference 

message: This article states 
:"M24 ... is contained within the symmetry group of the binary Golay code"
but the article Binary Golay code states
:"The automorphism group of the binary Golay code is the Mathieu group M_{23}".  
This appears to me to be inconsistent. Maproom (talk) 10:19, 25 June 2012 (UTC)
thread_title:  Contradiction 

message: Substituted at 02:19, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Read: ... also turn out ot be sporadic simple groups.
What is ot in this context? Jumpow (talk) 21:43, 1 December 2017 (UTC)
thread_title:  What is '''ot'''? 

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thread_title:  External links modified (January 2018) 

message: So Cameron (1999) lists all Mathieu groups except M22 as being 4-transitive. How can this be, when M24 and M12 are 5-transitive? I imagine it is because some of their subgroups are 4-transitive, but this doesn't necessarily make M24 and M12 4-transitive at the helm, does it? It's maybe simply a double transitivity that is then quite special for M24 and M12; however we also have M22 that is 3-transitive, while M11 is 4-transitive (sharply) (this makes sense since M11 is not a subgroup of M22, but M10 is a subgroup to both M22 and M11 with differing actions inside, respectively). I need to follow the construction and proof directly myself to double check, but these are somewhat complicated. Could anyone help me out in understanding this, I don't know if it is an error, and I would think otherwise beforehand. Also, are there other sources which might discuss this more directly, and in depth? I appreciate any feedback. Radlrb (talk) 16:58, 26 February 2023
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P.S. The only other relatively cryptic allusion I know that Cameron gives regarding 4/5-transitivity is from his 1992 paper, where he states on page 143:  I am going to try and find which old theorem of Jordan this is, and hopefully understand the underlying principles, and then input it into the article to give clarity. There is also this: Jordan’s theorem (multiply transitive groups) without going into detail. Radlrb (talk) 17:18, 26 February 2023 (UTC)
thread_title:  Transitive groups