Conway's group Co0 - Elements of orders 72, 80, 120?
- From: JEMebius <jemebius@xxxxxxxxx>
- Date: Mon, 26 May 2008 22:30:01 +0000 (UTC)
A question related to J.H.Conway's groups Co0 and Co1
-----------------------------------------------------
The group Co0 is the group of automorphisms of the Leech lattice which leave one lattice
point invariant. Is is a finite subgroup of the 24D rotation group SO(24). It is commonly
denoted by .0 (by its discoverer John Horton Conway) or by Co0.
Its centre Z2 consists of the non-rotation and the central inversion.
The factor group Co0/Z2 is a simple group, the first of Conway's sporadic simple groups.
It is commonly denoted by .1 or by Co1.
Co1 contains elements of orders 1 through 16, 18, 20, 21 through 24, 26, 28, 30, 33, 35,
36, 39, 40, 42, 60. (Reference: J.H.Conway et al.: ATLAS of Finite Groups)
Some time ago I explored the group Co0 by means of a specially designed exploration and
visualisation program (*).
I found 24D rotations of all the orders mentioned above and furthermore of orders 46, 52,
56, 66, 70, 78, 84.
Cyclic subgroups of Co0 of these orders necessarily contain the central inversion;
otherwise they would be mapped onto cyclic subgroups of Co1 of the same orders when
descending from Co0 to Co1.
One could also expect rotations of orders 72, 80, 120, but I never observed these orders.
My question: how to prove or disprove that Co0 contains elements of orders 72, 80, 120.
The answer might be found in Nicholas James Patterson's thesis "On Conway's group .0 and
some subgroups" (University of Cambridge, UK, 1973), but I hope that there is more readily
accessible literature on the fine structure of Co0.
Up to now I have only one speculation: if the Monster group contains a subgroup isomorphic
to Co0, then Co0 does not contain elements of order 120. This is because the highest order
found in the Monster is 119. (Reference: ATLAS)
Any help or hint would be much appreciated. Thanks in advance: Johan E. Mebius
(*)
------------------------------------------------------------------------------------------
http://www.xs4all.nl/~jemebius/Abconwayhelp.htm and
http://www.xs4all.nl/~jemebius/ABCONWAY.ZIP reachable from my home page at
http://www.xs4all.nl/~jemebius/index.html , item "downloads"
.
- Prev by Date: This Week's Finds in Mathematical Physics (Week 265)
- Next by Date: The General Infinite Exponential and Its Limit
- Previous by thread: This Week's Finds in Mathematical Physics (Week 265)
- Next by thread: The General Infinite Exponential and Its Limit
- Index(es):
Relevant Pages
|
|
