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The seven translates of ${0, 1, 3}$ form the Fano plane

Example

On the point set $Z / 7$ , consider the seven triples ${0, 1, 3}, {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 0}, {5, 6, 1}, {6, 0, 2},$ that is, the translates of ${0, 1, 3}$ modulo $7$ .

Facts & Assumptions

Given: The seven triples displayed above.

[L1]

A Steiner triple system is a $2$ - $(v, 3, 1)$ design (Steiner triple systems).

[L2]

A finite projective plane is a finite point-line incidence structure in which every two distinct points lie on exactly one line, every two distinct lines meet in exactly one point, every line contains at least three points, and there exist four points no three collinear (A finite projective plane).

Verification

technique · direct

1.1

Each displayed line has three points.

given

2.1

Let $T : = {0, 1, 3}$ . Its nonzero differences are $\pm 1, \pm 2, \pm 3$ , so every nonzero residue modulo $7$ occurs exactly once as $t_{2} - t_{1}$ with $t_{1}, t_{2} \in T$ . Therefore any two distinct points of $Z / 7$ lie in exactly one translate of $T$ . By [L1], the seven triples form a Steiner triple system on seven points.

step 1.1L1algebra

3.1

The same difference calculation shows that any two distinct displayed lines meet in exactly one point. The four points $0, 1, 2, 5$ contain no displayed line, so no three of them are collinear. Thus the same seven triples, read as lines, satisfy [L2]; every line has three points, so they form a finite projective plane of order $2$ . This projective plane is the Fano plane.

step 2.1L2algebra∎

Depends on

Used by

The complementary 2-(7,4,2) design is symmetric but not a projective plane Counterexample
The Fano plane has incidence matrix N with NN^T=2I+J Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §4.1 (standard reference, not scraped)

The seven translates of {0,1,3}{0,1,3} form the Fano plane