How small can such a group be?
in [Math]
|
Prev: Solving k^m = q mod N
Next: ever since 1842, the Doppler shift was assumed to exist for lightwaves and never experimentally verified Chapt 8 #138; ATOM TOTALITY
From: David Harden on 5 Jun 2010 14:05 Suppose G is a group and it contains subgroups isomorphic to A_8 and PSL(3,F_4). How small can G be? If I knew nothing about this problem, I would say we can take |G| <= 20160^2 = 406425600. However, I do know something about it: we can take |G| <= 10200960 because the Mathieu group M_23 works. But that is as far as I can go.
From: Derek Holt on 5 Jun 2010 16:57
On 5 June, 19:05, David Harden <roddl...(a)mit.edu> wrote: > Suppose G is a group and it contains subgroups isomorphic to A_8 and > PSL(3,F_4). How small can G be? > If I knew nothing about this problem, I would say we can take |G| <= > 20160^2 = 406425600. > However, I do know something about it: we can take |G| <= 10200960 > because the Mathieu group M_23 works. But that is as far as I can go. A_8 and PSL(3,4) must both be subgroups of some composition factor of G. If they are subgroups of different composition factors then |G| >=20160^2. It can be checked, by using the ATLAS for example, that M_23 is the smallest simple group that contains them both as subgroups, so M_23 is the (unique) smallest group containing them both. The Higman-Sims group HS of order 44352000 is the next smallest example. Derek Holt. |