Talk:Mathieu group

Constructions

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) The oldest reference I have actually seen is Carmichael's book. He does not attribute the generators to anyone. Scott Tillinghast, Houston TX 05:15, 27 June 2007 (UTC)

Internal links

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)

Language problem

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)

Introductory section has been revised. Scott Tillinghast, Houston TX (talk) 14:00, 24 March 2008 (UTC)

Summary of sources, aids to editors

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)

Multiply transitive phrasing

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 A_k-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 S_k and the alternating groups A_k+2

Cool dude ragnar (talk) 06:57, 18 May 2008 (UTC)

Sounds good. JackSchmidt (talk) 00:28, 19 May 2008 (UTC)

Fixed (again?). JackSchmidt (talk) 02:44, 16 May 2010 (UTC)

M21 is simple, just not a sporadic group. It is isomorphic to a classical group, namely PSL(3,4). It would take some rearranging to make the needed correction. Scott Tillinghast, Houston TX (talk) 23:27, 15 May 2010 (UTC)

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

Tables of elements of given order

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)

John McKay suggests 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)

Is M21 a Mathieu group?

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)

See the second paragraph of the intro. JackSchmidt (talk) 12:41, 23 July 2010 (UTC)

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)

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)

Request for a reference

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)

I've added a reference. Maproom (talk) 10:49, 4 March 2012 (UTC)

Contradiction

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)

What shape does this represent?

I know that the Monster Group represents a shape in 196.883 dimensions, which is obviously far too big to imagine. However with this group being so much smaller should assumingly relate to a much smaller shape. Considering we have pictures on Wikipedia of shapes up to 20 dimensions big and probably even bigger ones again, would it be possible add a picture or at least a description of what the shape's like on this wiki page? Robo37 (talk) 12:28, 9 May 2014 (UTC)

I'm not sure I understand you mean by "represent". But maybe this helps:

Colour the edges of an icosahedron as seen to the right. Consider the rotations of the icosahedron, which form the alternating group A5; and add in the operation of swapping the two balls at the ends of each yellow edge. These operations are all permutations on the set of 12 balls, and together they generate the group M12. I can find a reference for this.

Does this answer your question at all? Would it be a good thing to add to the article? If nothing else, it is (IMHO) pretty. Maproom (talk) 15:33, 9 May 2014 (UTC)

Thanks, that helped a lot actually. If you can find a good reliable reference I think it'll be a great addition to the page, although I'm not a professional mathematician so you're probably better off asking someone with a more educated opinion. If you're confident it's notable yourself mind, by all means add it. Robo37 (talk) 16:48, 11 May 2014 (UTC)

I find my recipe above for generating M12 using the icosahedron in the picture is wrong. This is correct:

Colour the edges of an icosahedron as seen to the right. Consider the rotations of the icosahedron, which form the alternating group A5; and add in the operation of cycling clockwise each of the four sets of three vertices connected by red edges. These operations are all permutations on the set of 12 balls, and together they generate the group M12.

I am still searching for a reference. Maproom (talk) 13:40, 12 May 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Mathieu 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.

This article is too technical. It needs to lead the typical reader to the importance and interest more directly. Geometry guy 23:51, 14 April 2007 (UTC)

Last edited at 23:51, 14 April 2007 (UTC). Substituted at 02:19, 5 May 2016 (UTC)

What is ot?

Read: ... also turn out ot be sporadic simple groups.

What is ot in this context? Jumpow (talk) 21:43, 1 December 2017 (UTC)

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Transitive groups

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

• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, p. 110, ISBN 978-0-521-65378-7
• Cameron, Peter J. (1992), Projective and Polar Spaces (PDF), University of London, Queen Mary and Westfield College, p. 143, ISBN 978-0-902-48012-4, S2CID 115302359

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: But N, a normal subgroup of a 5-transitive group, is at least 4-transitive, by an old theorem of Jordan... 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)