Talk:Conway group

message: I re-instated version 2 alongside version 3. The nice thing about version 2 is that lists of maximal subgroups have links to pages on the simple groups that are involved. Scott Tillinghast, Houston TX (talk) 21:45, 4 May 2008 (UTC)
thread_title:  Versions of Robert A. Wilson website 

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. I would welcome suggestions about what else I should include.
===Opening of article===
I added in some history, which I mainly got from Thomas Thompson's Carus monograph. He talks about the groups he has space to give adequate mathematical treatment. That includes the 3 Conway groups and some basics about M24, but he barely mentions the other 4 soradic groups of the second generation.
===Other sporadic groups===
The best source is Robert Griess's Twelve Sporadic Groups. Quite a bit of information, but there are many interesting questions he does not touch. He gives very little history on these groups.
For history I have resorted to the book edited by Brauer and Shih, a compilation of lectures from May 1968 conference at Harvard.
I am trying to enumerate all the conjugacy classes of simple subgroups and subquotients in Co0, but this may constitute original research.
===The group 212:M24===
From Thomas Thompson. He relates how it began Conway's investigation, how he managed to prove this is a proper but maximal subgroup of Co0. He could then deduce |Co0|.
===Maximal subgroups of various subgroups===
Mainly from Robert A. Wilson's Atlas website. Perhaps it would be better to list the maximal subgroups of Co0, but I don't know of a listing. The interesting thing about them is that some of them are direct products with maximal subgroups of Co1, and others are stem extensions. It is Co0 that is represented in SL(Q,24), as well as Co2 and Co3, but not Co1.
Scott Tillinghast, Houston TX (talk) 04:05, 9 June 2008 (UTC)
thread_title:  Summary of sources, aids to editors 

message: Mainly from Robert A. Wilson's Atlas website. Perhaps it would be better to list the maximal subgroups of Co0, but I don't know of a listing. The interesting thing about them is that some of them are direct products with maximal subgroups of Co1, and others are stem extensions. It is Co0 that is represented in SL(Q,24), as well as Co2 and Co3, but not Co1.
Scott Tillinghast, Houston TX (talk) 04:05, 9 June 2008 (UTC)
thread_title: Maximal subgroups of various subgroups

message: Scott Tillinghast, Houston TX (talk), at WT:WPM#Separate page, suggested separate articles on the McLaughlin and Suzuki sporadic simple groups. Suzuki sporadic group and McLaughlin group (mathematics) already exist as redirects, but on his behalf I've put some split tags up for discussion. -- Radagast3 (talk) 14:22, 16 April 2010 (UTC)
replies: text: Personally, I see no problem with turning the redirects into articles, copying text from here to begin with. That would need to be noted in the initial edit summary, of course. -- Radagast3 (talk) 08:09, 19 April 2010 (UTC)
thread_title:  Possible split 

message: Figure of speech for something seemingly unrelated. What I was saying was that the Conway's work unified some previously known sporadic groups that seemed unrelated. Namely Higman-Sims, Hall-Janko, Suzuki, and McLaughlin. Scott Tillinghast, Houston TX (talk) 23:23, 4 November 2010 (UTC)
thread_title:  Rabbit trails 

message: Perhaps a proof should be included that all involutions of the same trace in Co0 are conjugate. A useful tool: if the product of 2 involutions has odd order n, these involutions are conjugate in a dihedral subgropu of order 2n. Scott Tillinghast, Houston TX (talk) 19:27, 18 January 2016 (UTC)
replies: text: {\mathbf 1/2}  \left ( \begin{matrix}
1 &  -1 & -1 & -1 \\
-1 &  1 & -1 & -1 \\
-1 & -1 &  1 & -1 \\
-1 & -1 & -1 & 1 \end{matrix} \right )
has with the diagonal matrix {-1,1,1,1} a product of order 3. Hence all matrices of type ζ (see article) are conjugate to diagonal matrices. Scott Tillinghast, Houston TX (talk) 20:18, 18 January 2016 (UTC)
thread_title:  Involutions 

message: Co0 has a subgroup isomorphic to S3 permuting 3 subspaces of the Leech lattice, each of dimension 8. That led to the discovery of the Suzuki chain. Let Z be the center of Co0 (order 2) and C0 be the centralizer of the S3. Griess proves that C0 is the double cover of A9 but is not clear about details. He does not prove that C0/Z is simple but perhaps this is similar to proving Co1 is simple. Let t be a diagonal involution in C0. It is not so difficult to show that the centralizer of {±t} in C0/Z has form 23:S4. A survey of simple groups shows only 2 possibilities with an involution with such a centralizer: A8 and A9. Griess is obscure about why a Sylow 3-subgroup of C0 is larger than 32. Because C0 contains a subgroup isomorphic to SL(2,7) it is 2.A9, not the direct product 2xA9. Scott Tillinghast, Houston TX (talk) 15:36, 23 January 2018 (UTC)
thread_title:  Centralizer of an S<sub>3</sub>