nLab Fano plane

Idea

The Fano plane is the projective plane over the finite field $\mathbf{F}_2$ with two elements. It is denoted either $\mathbb{F}_2\mathbb{P}^2$, $\mathbb{P}^2_{\mathbb{F}_2}$ (in algebraic geometry), or $PG(2,2)$ (in finite geometry). It is often pictured as at right, but there are many different representations.

Properties

Automorphisms

The automorphism group $PGL(3,\mathbb{F}_2)$ is isomorphic to the projective special linear group $PSL(2,\mathbb{F}_7)$.

Related concepts

• octonion

References

• Wikipedia, Fano plane

• John Baez, The Fano plane