Editing Steiner system (section)

== The Steiner system S(5, 6, 12) ==
There is a unique S(5,6,12) Steiner system; its automorphism group is the [[Mathieu group]] M<sub>12</sub>, and in that context it is denoted by W<sub>12</sub>.

=== Projective line construction ===
This construction is due to Carmichael (1937).<ref>{{harvnb|Carmichael|1956|page=431}}</ref>

Add a new element, call it {{mvar|∞}}, to the 11 elements of the [[finite field]] {{mvar|'''F'''}}<sub>11</sub> (that is, the integers mod 11). This set, {{mvar|''S''}}, of 12 elements can be formally identified with the points of the [[projective line]] over {{mvar|'''F'''}}<sub>11</sub>. Call the following specific subset of size 6,
:<math>\{\infty,1,3,4,5,9\}, </math>
a "block" (it contains {{math|∞}} together with the 5 nonzero squares in {{mvar|'''F'''}}<sub>11</sub>).  From this block, we obtain the other blocks of the {{mvar|S}}(5,6,12) system by repeatedly applying the [[linear fractional transformation]]s:
:<math>z' = f(z) = \frac{az + b}{cz + d},</math>
where {{mvar|a,b,c,d}} are in {{mvar|'''F'''}}<sub>11</sub> and {{math|1= ''ad &minus; bc'' = 1}}.
With the usual conventions of defining {{math|1= ''f'' (&minus;''d''/''c'') = ∞}} and {{math|1= ''f'' (∞) = ''a''/''c''}}, these functions map the set {{mvar|''S''}} onto itself. In geometric language, they are [[Projectivity|projectivities]] of the projective line. They form a [[group (mathematics)|group]] under composition which is the [[projective special linear group]] {{mvar|PSL}}(2,11) of order 660. There are exactly five elements of this group that leave the starting block fixed setwise,<ref>{{harvnb|Beth|Jungnickel|Lenz|1986|page=196}}</ref> namely those such that {{math|1= ''b=c=0''}} and {{math|1= ''ad''=1}} so that {{math|1= ''f(z) = a''<sup>2</sup> ''z''}}. So there will be 660/5 = 132 images of that block. As a consequence of the multiply transitive property of this group [[Group action (mathematics)|acting]] on this set, any subset of five elements of {{mvar|''S''}} will appear in exactly one of these 132 images of size six.

=== Kitten construction ===
An alternative construction of W<sub>12</sub> is obtained by use of the 'kitten' of R.T. Curtis,<ref>{{Harvnb|Curtis|1984}}</ref> which was intended as a "hand calculator" to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid of numbers, which represent an [[affine geometry]] on the [[vector space]] F<sub>3</sub>xF<sub>3</sub>, an S(2,3,9) system.

=== Construction from K<sub>6</sub> graph factorization ===
The relations between the [[graph factorization|graph factors]] of the [[complete graph|complete graph K<sub>6</sub>]] generate an S(5,6,12).<ref>{{cite web| url = http://linear.ups.edu/eagts/section-24.html| title = EAGTS textbook}}</ref> A K<sub>6</sub> graph has 6 vertices, 15 edges, 15 [[perfect matching]]s, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices  (labeled 123456) and the set of factorizations (labeled ''ABCDEF'') provide one block each. Every pair of factorizations has exactly one perfect matching in common. Suppose factorizations ''A'' and ''B'' have the common matching with edges 12, 34 and 56. Add three new blocks ''AB''3456, 12''AB''56, and 1234''AB'', replacing each edge in the common matching with the factorization labels in turn. Similarly add three more blocks 12''CDEF'', 34''CDEF'', and 56''CDEF'', replacing the factorization labels by the corresponding edge labels of the common matching. Do this for all 15 pairs of factorizations to add 90 new blocks. Finally, take the full set of <math>\tbinom{12}{6} = 924 </math> combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far. Exactly 40 blocks remain, resulting in {{nowrap|1=2 + 90 + 40 = 132}} blocks of the S(5,6,12). This method works because there is an [[automorphisms of the symmetric and alternating groups#The exceptional outer automorphism of S6|outer automorphism on the symmetric group ''S''<sub>6</sub>]], which maps the vertices to factorizations and the edges to partitions. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism.