 Derangement in a transitive group action

A derangement is an element fixing none of the points of a group action. Every finite transitive action on at least two points has a derangement. Count pairs $(g,x)$ with $gx=x$: transitivity gives ${\sum }_g|\mathrm{Fix}(g)|={\sum }_x|G_x|=|G|$. If every element fixed a point, the identity's extra fixed points would make this sum strictly larger than $|G|$. This is the Burnside lemma counting argument specialized to existence of a derangement.