Talk:Conway group

Versions of Robert A. Wilson website

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)

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. 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)

I just checked out a new book by Robert A. Wilson: 'The Finite Simple Groups.' He lists all the maximal subgroups of the 3 Conway groups and some maximal subgroups of Co0. One topic of interest is a chain of subgroups involving the Suzuki group at one end and the alternating group A9 at the other. I am drafting a section about this chain. Scott Tillinghast, Houston TX (talk) 23:12, 1 April 2010 (UTC)

Possible split

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)

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)

I see that the new articles have been created. I shall be finding things to add on the McLaughlin and sporadic Suzuki groups. I would also like to see an article on the infinite family called the Suzuki groups. They are of interest as the only simple groups of composite order not divisible by 3. They have little to do with the sporadic Suzuki group. Scott Tillinghast, Houston TX (talk) 07:59, 7 November 2010 (UTC)

Rabbit trails

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)

Compatibility

The only matrices specified in this article are those of type ζ, introduced as non-monomial matrices. They are built from a 4x4 matrix η and its negative.

The standard subgroup of type 2.A9 in Co0 includes the non-monomial matrix η, -η, η, -η, η, -η. The diagonal dodecads of 2⁴ can be identified with some of the 70 4-subsets of a set of 8. PSL(2,7) (degree 8) splits those 4-subsets into 2 orbits of 14 and an orbit of 42. All 3 orbits include complements. The 2 orbits of 14 both generate groups of order 16 under symmetric difference. The orbit of 42 does not generate a good group. What representation of 2⁴:PSL(2,7) will be compatible, in the sense of generating a subgroup of Co0?

The ζ matrix and 2⁴:PSL(2,7) generate a block sum of 3 identical 8x8 matrix groups. I do not know whether this is 2.A8 or the full group 2.A9. Scott Tillinghast, Houston TX (talk) 23:10, 23 December 2015 (UTC)

The section on the monomial subgroup really ought to define precisely a standard representation of the binary Golay code, thus one of M24. What would be an easiest way to do this? Scott Tillinghast, Houston TX (talk) 21:30, 24 December 2015 (UTC)

The group PSL (2,7) can be represented as the linear fractional group on F₇ ∪ {∞}, generated by permutations (0123456) and (0∞)(16)(23)(45). Griess (p. 59) uses the labeling:

∞ 0 |∞ 0 |∞ 0

3 2 |3 2 |3 2

5 1 |5 1 |5 1

6 4 |6 4 |6 4

One dodecad can be represented either by the subset {0,1,3,∞} or by its negative {0,4,6,∞}. Scott Tillinghast, Houston TX (talk) 02:13, 26 December 2015 (UTC)

I have defined a convenient representation in the article on the binary Golay code and made a link. Scott Tillinghast, Houston TX (talk) 17:15, 27 December 2015 (UTC)

Involutions

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)

The matrix η =

${\mathbf {1} /2}\left({\begin{matrix}1&-1&-1&-1\\-1&1&-1&-1\\-1&-1&1&-1\\-1&-1&-1&1\end{matrix}}\right)$

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)

A permutation matrix representing a product of transpositions is easily diagonalized in SL(24,R) but this requires sqrt 2. Diagonalizing in Co₀ is more complicated. Scott Tillinghast, Houston TX (talk) 03:18, 21 January 2016 (UTC)

Centralizer of an S₃

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)