Talk:Higman–Sims group

Presentation

Is there any value to having the presentation for HS on this page? It has been copied from the Atlas (http://brauer.maths.qmul.ac.uk/Atlas/spor/HS/) and is hopeless (because completely unmotivated) as a definition. Unless I hear strong objections, I shall remove it.

I have removed it. The previous content was:

HS can also be defined in terms of generators a and b and the following relations:

$a^{2}=b^{5}=(ab)^{11}=(ab^{2})^{10}=[a,b]^{5}=[a,b^{2}]^{6}=[a,bab]^{3}=$

$ababab^{2}ab^{-1}ab^{-2}ab^{-1}ab^{2}abab(ab^{-2})^{4}=$

$ab(ab^{2}(ab^{-2})^{2})^{2}ab^{2}abab^{2}(ab^{-1}ab^{2})^{2}=$

$abab(ab^{2})^{2}ab(ab^{-1})^{2}ab(ab^{2})^{2}ababab^{-2}ab^{-1}ab^{-2}=1.$

--Huppybanny 09:48, 27 October 2005 (UTC)

Representation

The Mathieu group M₂₂ occurs as a maximal subgroup of HS. One permutation matrix representation of M₂₂ fixes a 2-3-3 triangle with vertices (0²⁴), (5,1²³), and (1,5,1²²). A non-monomial matrix is needed to complete generation of a representation of HS. Scott Tillinghast, Houston TX (talk) 21:28, 21 December 2015 (UTC)

Wilson (2009) gives an example of a Higman-Sims graph within the Leech lattice, permuted by the representation of M₂₂ above:

• 22 points of shape (1,1,-3,1²¹)
• 77 points of shape (2,2,2⁶,0¹⁶)
• A 100th point (4,4,0²²)

Differences of adjacent points are of type 3; those of non-adjacent ones are of type 2. Scott Tillinghast, Houston TX (talk) 16:08, 23 December 2015 (UTC)

Problem: find an involution fixing the aforesaid 2-3-3 triangle and transposing all 100 points of the aforesaid graph. What is its trace? Co₃ contains involutions of trace 0 but not of trace -8. Scott Tillinghast, Houston TX (talk) 09:07, 29 December 2015 (UTC)

A 2-3-3 triangle?

Not if the origin is one vertex. Rather a translation of a 2-3-3 triangle. It would define the same vector space over the rationals as the original triangle, but not the same Z-module. Scott Tillinghast, Houston TX (talk) 15:56, 15 January 2016 (UTC)

Disregard. Scott Tillinghast, Houston TX (talk) 04:02, 25 January 2018 (UTC)

Format problem

For some reason my new table of conjugacy classes goes to the end of the page. Scott Tillinghast, Houston TX (talk) 21:25, 24 January 2018 (UTC)

Corrected! Changed |-} to |} at end.

Stabilizer of edge

An involution class 2A (trace 8) transposes 40 non-edges, fixes 20 vertices in the Higman-Sims graph.

The stabilizer of an edge includes the group M₂₁ = PSL(3,4), fixing the 100th vertix C and one of the 22 points. It seems the full stabilizer PSL(3,4):2 of that edge is generated by an involution class 2B (trace 0) which would transpose 50 edges. Scott Tillinghast, Houston TX (talk) 05:53, 2 June 2019 (UTC)

I have not found any matrix construction of a 2B involution in the literature. It would exchange the 100th vertex C with one of the 22 points, could exchange the other 21 points with hexads. There would be 56 hexads left over. Scott Tillinghast, Houston TX (talk) 18:14, 2 June 2019 (UTC)

A class 2B matrix gives a bonus: with M₂₃ a representation of Co₃. Scott Tillinghast, Houston TX (talk) 22:56, 2 June 2019 (UTC)

An involution class 2A can only transpose non-edges, because it is conjugate to one in the M₂₂ that exchanges hexads only with hexads. A transposed pair of adjacenr (disjoint) hexads would imply a dodecad whose complement includes an octad. Scott Tillinghast, Houston TX (talk) 12:48, 4 June 2019 (UTC)

But Wilson (2009) [p. 213] constructs a monomial class 2B matrix within a representation of the maximal subgroup 2⁴S₆, stabilizer of a non-edge. This involution transposes a non-edge. Scott Tillinghast, Houston TX (talk) 03:31, 9 June 2019 (UTC)

The orbits of M₂₁ are 1 (point 1), 1 (100th vertix), 21 (points), 21 (hexads containing point 1), 56 (hyper-ovals). The 2 fixed points form an edge. Magliveras may be wrong that the normalizer M₂₁:2 has orbits 2, 42, 56. Scott Tillinghast, Houston TX (talk) 20:22, 10 June 2019 (UTC)

Perhaps the edge is transposed by an element of order 4. In that case a stabilizer M₂₁.2 would not be a split extension. Scott Tillinghast, Houston TX (talk) 17:10, 16 June 2019 (UTC)