From: magidin@hilbert.berkeley.edu (Arturo Magidin)
Subject: Re: Steiner system
Date: 18 Feb 1999 15:13:26 GMT
Newsgroups: sci.math
Keywords: Steiner systems: what they are, example of construction

In article <7aged2$ld8$1@nnrp1.dejanews.com>,
 <irine_c_p@my-dejanews.com> wrote:
>what's Steiner system?	i vaguely remeber that it has something to do with a
>system of q^2+q+1 points and lines, 2 points determine one line and (q+1)
>lines intersect at one point.  can someone confirm this?  is there anything
>called projective Steiner system? if there is, what is it?

A Steiner system of type (p,q,r) is a collection of subsets of a set
with p elements, each subset containing q elements, and such that
given any r elements of the set, there exist exactly one subset in the
collection which contains all r elements.

One possibility is obtained over a finite field of q elements, where
the plane consists of q^2 elements, each line contains q elements, and
any 2 points determine a unique line containing them; hence the affine
plane over a field of q elements is of type (q^2,q,2). You can do
something similar with projective space over a field of q elements.

One source on abstract Steiner Systems is J.J. Rotman's "Theory of
Groups", 4th Edition.

======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
==============================================================================

From: Robin Chapman <rchapman@mpce.mq.edu.au>
Subject: Re: The odd number 6
Date: Fri, 26 Feb 1999 09:01:47 +1100
Newsgroups: sci.math

torquemada@my-dejanews.com wrote:
> 
> In article <slrn7bl0qc.8ck.huw@eryr.adar.net>,
>   huw@eryr.adar.net (Huw Davies) wrote:
> 
> > I think it goes the other way! you can use the outer automorphism of S_6 to
> > construct the (5,6,12) Steiner system, and hence M_12...
> 
> Hmmm...I've not yet taken this particular route to S(5,8,24). Where can I find
> this written up? (Please bear in mind I no longer have easy access to an
> academic library so papers are out. A book or online reference is better).

The first stage of this (S_6 to S(5,6,12)) is quite easy. Take an outer
automorphism phi:S_6 -> S_6 but let's label the elements permuted in the
first S_6 as 1,2,3,4,5,6 and the second S_6 as a,b,c,d,e,f. We construct
an S(5,6,12) on {1,2,3,4,5,6,a,b,c,d,e,f} as follows. {1,2,3,4,5,6}
and {a,b,c,d,e,f} are hexads. If we have a transposition (jk) in the first
S_6 then phi((jk)) = (uv)(wx)(yz) . Then {j,k,u,v,w,x} etc., and their
complements are hexads. Also if phi((ijk)) = (uvw)(xyz) then {i,j,k,u,v,w}
are hexads.

For M_12 to S(5,8,24) I'll be briefer. Let psi:M_12 -> M_12 be the outer
automorphism. Again think of the M_12s as acting on disjoint index sets
and take our 24-set to be the union of these. Each point stabilizer is
mapped to a transitive subgroup by psi. The intersection of two point
stabilizers is mapped to an imprimitve subgroup of M_12 having
two blocks of six points. The two points plus either of the blocks forms
an octad, and these generate all octads in the usual way.

-- 
Robin Chapman				+ "Going to the chemist in
Department of Mathematics, DICS		-  Australia can be more
Macquarie University			+  exciting than going to
NSW 2109, Australia			-  a nightclub in Wales."
rchapman@mpce.mq.edu.au			+   Howard Jacobson,
http://www.maths.ex.ac.uk/~rjc/rjc.html	-     In the Land of Oz