# Fano postulate

A proposition of projective geometry established by G. Fano (1892). It consists in the fact that the diagonal points of a quadrangle are not collinear. The Fano postulate is equivalent to the fact that the characteristic of the skew-field $K$ associated with the projective geometry in question is not equal to 2. The Fano postulate does not hold, for example, in the finite projective plane consisting of seven points and lines associated with the skew-field $K$ of the two elements 0 and 1.

#### Comments

On the other hand, it was shown by A.M. Gleason [a2] that a finite projective plane in which the diagonal points of any quadrangle are collinear, is a projective plane over a field (of characteristic 2).

#### References

[a1] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) Zbl 0095.34502 |

[a2] | A.M. Gleason, "Finite Fano planes" Amer. J. Math. , 78 (1956) pp. 797–807 Zbl 0072.38001 |

**How to Cite This Entry:**

Fano postulate.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fano_postulate&oldid=40965