Steiner Triple Systems

Definition 5.8   A Steiner Triple System (STS) is a hypergraph $\cal H$ (in which the edges are called lines) such that

(a)

every line consists of exactly 3 points(=vertices) (i.e., $H$ is 3-uniform);

(b)

every pair of (distinct) points is contained in a unique line.

In other words, a STS satisfies condition (C1) above and is maximal with that property: The number of edges is exactly equal to the upper bound obtained above: $\vert E\vert=n(n-1)/6$, where $n$ is the number of vertices. In particular we must have that $n(n-1)$ is divisible by 6.

This necessary condition on $n$ also turns out to be sufficient.

We construct a STS $\cal F$ on 7 vertices (with 7 edges) as follows: Take the vertices of a(n equilateral) triangle along with the bisectors of the edges and the (in)center of the triangle. Let the edges of the STS consist of the following:

(a)

the 3 vertices on each edge of the triangle (for a total of 3 edges).

(b)

the 3 vertices on each angle bisector.

(c)

the 3 vertices on the (natural) inscribed circle.

This configuration is called the ``Fano Plane'' see Figure .

Figure: The Fano Plane

52#1

The incidence matrix for the Fano plane is as follows (the rows correspond to lines, columns to points, ``1" indicates incidence):

	0	1	2	3	4	5	6
a	1	1	0	1	0	0	0
b	0	1	1	0	1	0	0
c	0	0	1	1	0	1	0
d	0	0	0	1	1	0	1
e	1	0	0	0	1	1	0
f	0	1	0	0	0	1	1
g	1	0	1	0	0	0	1

53#2

The group of automorphisms of the Fano plane is a very important group: it is the second smallest finite simple group. ``Simple groups'' to ``groups'' are like ``atoms'' to ``chemical compounds,'' so according to this analogy, the automorphism group of the Fano plane is the ``Helium of group theory.''

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