Talk:Conway group Co1

I am inclined to put the section on the monomial subgroup first, right after the opening section, for historical reasons if nothing else. Conway began his investigation with this subgroup and quickly found the order of Conway-0. Scott Tillinghast, Houston TX (talk) 01:14, 28 July 2014 (UTC)

Co₀ or Co₁?

I am attracted to putting the focus on Co₀, as a matrix group. One can often avoid writing out 24-by-24 matrices by speaking in terms of their symmetry properties.

Thomas Thompson's book describes a generating matrix outside the monomial group, but the exposition is not easy, and there appear to be some errors in signs. I would find it easier to describe the matrix as an adjunction of 6 4-by-4 matrices. The basic matrix is

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

It is an involution. The full matrix is formed by adjoining odd numbers of copies and their negatives. Thompson neglects to use negatives and his matrix produces vectors that cannot be in the Leech lattice. Incidentally, this construction makes use of sextets (described under the M24 article).

Co₀ should be of interest as providing representatations of the many signifiant subgroups it contains. Many may be reducible to a dimension lower than 24. I would to see matrices of order 9 and 13.

I have not found a source for a list of maximal subgroups of Co₀. Some are direct products and some are double covers. Scott Tillinghast, Houston TX (talk) 16:50, 29 July 2014 (UTC)

Robert Griess's book (p. 97) correctly describes a certain non-monomial matrix as a block sum of 6 4-by-4 matrices. Scott Tillinghast, Houston TX (talk) 18:22, 30 July 2014 (UTC)

A question of representations

I have found it convenient to represent the Leech lattice on the smallest scale for which all co-ordinates are integers. That gives a natural representation of Co₀ in which a maximal subgroup is composed of integer matrices. I doubt there is a 24-dimensional representation composed entirely of integer matrices. Does anyone want to use a representation with only permutation matrices? A look at a character table should answer whether a 24-dimensional representation over Z exists. Scott Tillinghast, Houston TX (talk) 16:40, 2 August 2014 (UTC)

What's next?

I have presented a way to generate a representation of the Leech lattice and its corresponding representation of Co₀. There are some tedious details of proof that I prefer to leave to a reader who really wants to work them out.

Co₀ is said to have 4 conjugacy classes of involutions and Co₁ to have 3. Any involution in Co₀ is said to be conjugate to a codeword in the Golay code, but I do not yet understand Griess's proof. In any event, every involution is conjugate to a monomial involution, because the monomial group contains Sylow 2-subgroups. One class in Co₁ lifts to elements of order 4 in Co₀. Perhaps that class is represented by a block sum of 12 copies of

$\left({\begin{matrix}0&-1\\1&0\end{matrix}}\right)$

The article might also describe centralizers of involutions.

M₂₄ and thus the monomial subgroup have 2 classes of 3-elements, but there 4 classes in Co₀ and Co₁. One type of monomial 3-element, corresponding to a permutation of shape 3¹², leads to the Suzuki chain. The centralizer of such a 3-element is 3 x 2.A₉. It would include a matrix of order 9, which could not be monomial. This group may be easier to generate with a matrix whose square is the central involution, conjugate to the block sum I just suggested, but not monomial.

Several types of Co₀ elements could be discussed in other articles about subgroups of the Conway groups. Scott Tillinghast, Houston TX (talk) 04:20, 3 August 2014 (UTC)

A standard permutation matrix representation of M₂₃ fixes a point of shape (-3,1²³), type 2, and a point of shape (5,1²³), type 3. M₂₃ is maximal both in Co₂ and Co₃. The direct products 2xCo₂ and 2xCo₃ are both maximal in Co₀. One generator expands M₂₃ to a nice representation of Co₂ or Co₃. Scott Tillinghast, Houston TX (talk) 20:44, 20 December 2015 (UTC)

A block sum of η =

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

and 5 copies of its negative fixes the vector (-3,1²³) but not (5,1²³). Hence Co₂ has a convenient representation inside the standard representation of Co₀, generated by this matrix along with the M₂₃ fixing the first coordinate. Scott Tillinghast, Houston TX (talk) 00:04, 21 December 2015 (UTC)

Constructing the group 3 x 2.A₉

I have not found a clear proof in the literature that the centralizer of one type of 3-element has the form 3 x 2.A₉. Griess offers a cryptic proof (p. 116). Conway in Conway/Sloane (p. 270) offers some tantalizing hints.

A₈ (as well as A₉) has 2 types of simple subgroups of order 168. I will refer to those of degree 7 as GL(3,2) and those of degree 8 as PSL(3,7).

In M₂₄ a permutation t of shape 3⁸ has as its centralizer the direct product <t> x PSL(2,7). That group operates on a trio of octads. Griess also finds that t commutes with a diagonal group U of type 2⁴; PSL(2,7) is transitive on 14 of these involutions, which are dodecads. The rest of U is the center C of Co₀. Thus the centralizer of t in Co0 includes a group K of type U:PSL(2,7). At this point Griess gets cryptic and I will proceed on my own.

The group U must permute 8 conjugates of PSL(2,7) in K. Each PSL(2,7) permutes its other 7 conjugates. Hence there is a homomorphism of K onto 2³:GL(3,2) and its kernel is C. It follows that K has an embedding in the double cover 2.A₈. In the latter the PSL(2,7) lift to a direct product 2 x PSL(2,7) but the GL(3,2) lift to a double cover isomorphic to SL(2,7). That is because double transpositions go to 4-elements but quadruple transpositions remain involutions.

I suspect that K is the maximum intersection of the monomial group with a 2.A₉. Now it remains to show that 2.A₉ is the full centralizer of <t>. A matrix of order 9 would generate 2.A₉.

Conway remarks (again p. 270) it is a miracle that 2³:GL(3,2) has 2 types of subgroups of order 168 that act as normalizers on the normal subgroup 2³. An interesting consequence of the exceptional isomorphism between GL(3,2) and PSL(2,7). — Preceding unsigned comment added by Scott Tillinghast, Houston TX (talk • contribs) 15:13, 2 September 2014 (UTC)

The group U:PSL(2,7) contains subgroups of order 56. Consider action on just one octad. There is a convenient representation with essentially dimension 7, generated by a cyclic permutation of the 7 coordinates and a diagonal matrix with 4 -1's on the diagonal. The group of order 56 is a starting point for a simple group of order 504, whose Schur multiplier is trivial. Pick a convenient generator of order 9. Scott Tillinghast, Houston TX (talk) 21:05, 18 December 2015 (UTC)

Explain the word shape

The word shape is used in what appears to be a technical sense, including a notation like 0^23, but there is no explanation or a reference. Conway's and Sloane's book routinely uses this term but does not explain it anywhere. It does not appear in Wilson's "Graduate Texts" book. Could any of the authors indicate from which source they obtained it, or better still, which of their sources explains this technical use of the word shape?--Lieven Smits (talk) 16:41, 22 April 2015 (UTC)

Transfer of sections?

Perhaps the sections on the monomial group, involutions, and the Suzuki chain would better fit into the article 'Conway group.' These sections relate more to Co₀ than to Co₁. — Preceding unsigned comment added by Scott Tillinghast, Houston TX (talk • contribs) 19:54, 22 December 2015 (UTC)

It seems logical to put all references to matrices into the article 'Conway group' or into those about subgroups Conway 2, Conway 3, Higman-Sims, or McLaughlin. Conway 1 is an abstract group. Scott Tillinghast, Houston TX (talk) 15:54, 23 December 2015 (UTC)