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A projective plane of order $n$ is a symmetric $2$ - $(n^{2} + n + 1, n + 1, 1)$ design

Statement

Let $(P, L)$ be a finite projective plane of order $n$ . Then, with points as the ground set and lines as the blocks, $(P, L)$ is a symmetric $2$ - $(n^{2} + n + 1, n + 1, 1)$ design.

Facts & Assumptions

Given: A finite projective plane $(P, L)$ of order $n$ .

[L1]

Every line has exactly $n + 1$ points (The order of a finite projective plane).

[L2]

Any two distinct points lie on exactly one line (A finite projective plane).

[L3]

A plane of order $n$ has $n^{2} + n + 1$ points and the same number of lines (A finite projective plane of order $n$ has $n^{2} + n + 1$ points and the same number of lines).

Proof

technique · direct

1.1

By [L1], every block has size $n + 1$ , and by [L2], every pair of distinct points lies in exactly one block. Thus $(P, L)$ is a $2$ - $(n^{2} + n + 1, n + 1, 1)$ design.

L1L2L3

2.1

By [L3], the number of blocks equals the number of points, namely $n^{2} + n + 1$ , so the design is symmetric.

L3∎

Depends on

Used by

The complementary 2-(7,4,2) design is symmetric but not a projective plane Counterexample

Dependency tree · two levels

6 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

Noam D. Elkies, Math 155 notes: Feb. 3 (standard reference, not scraped)

A projective plane of order nn is a symmetric 22-(n2+n+1,n+1,1)(n2+n+1,n+1,1) design