Editing Steiner system (section)

=== 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.