# Projective line construction of exceptional Steiner system.

Wikipedia gives a beautiful construction of the $$S(5,6,12)$$ Steiner system; take as blocks the subset of the projective line over the field with $$11$$ elements consisting of $$\{\infty,1,3,4,5,9\}$$ and all its images under the natural action of $$PSL_2(11)$$.

I am hoping to find a slick verification that this does indeed form such a Steiner system, that is, that any $$5$$ elements are contained in a unique one of the above blocks. Here are my thoughts so far: note that the 6 elements in our starter block are just the non zero squares and $$\infty$$, so that it is stabilized in $$PSL_2(11)$$ by at multiplication by squares. As $$PSL_2(11)$$ has order $$660$$, this shows there are at most $$660/5 = 132$$ blocks. In fact the stabilizer of a block has size only $$5$$, which I can verify but in a slightly computational way. If one does that, it suffices to show no two distinct blocks intersect in $$5$$. Another helpful fact is that $$PGL_2(11)$$ acts $$3$$ transitively on the $$12$$ points of the projective line, and $$PSL_2(11)$$ can at least map any ordered $$3$$ to any other unordered $$3$$ (in $$3$$ ways), which seems helpful, but I can't quite show what I want from this. Wikipedia gives a reference to this statement, but it doesn't contain a proof.

I care about this result as I know a handful of constructions of $$S(5,6,12)$$ already, as well as its uniqueness, thus understanding result will further shed light on this Steiner system's symmetries, which form the sporadic group $$M_{12}$$.