From: jamesrheckman@gateway.netnospam (Jim Heckman)
Subject: Re: The Monster and the Leech
Date: 01 Oct 2000 04:05:31 GMT
Newsgroups: sci.physics.research
Summary: [missing]

>From: jamesrheckman@gateway.netnospam  (Jim Heckman)
>Date: 9/29/00 9:06 PM Pacific Daylight Time
>Message-id: <20000930000618.18144.00000214@ng-bd1.news.gateway.net>

> [...]

>But as I said above, I can't see where we've actually established that
>Co_0 is the covering group of Co_1,

Aha! I've now found a reference that verifies that Co_0, the full
orthogonal automorphism group of the Leech lattice, is indeed the
(universal) covering group of the sporadic simple group Co_1 via its
Schur multiplier Z_2. (The center of Co_0 as Aut(Leech) is just {I, -I}
in SO(24,R).) This confirms that Co_0=Aut(Leech) lies entirely in
SO(24,R), i.e. that it's a pure rotation group, with no det = -1 elements,
since it can't have an index-2 subgroup.

>as opposed to the other possibility
>Co_0 = (Co_1 x Z_2). (There are no other possibilities with order
>2|Co_1| and Co_1 as a composition factor, since Out(Co_1) is trivial.)

FWIW, I'm no longer so sure these are the only two possibilities. I was
generalizing from small alternating groups A_n, which have lots of
subgroups of small index that intersect each other and allow one to
conclude that the only extensions of A_n by Z_2 are S_n and
(A_n x Z_2) -- but that argument breaks down once the subgroups'
indices get big enough that they don't have to intersect.

If the result *is* true, I suspect it can be proved by looking closely at
the normalizers of Co_1's Sylow p-subgroups for odd p...

--
Jim Heckman