= Derangement in a transitive group action
{title2=$\operatorname{Fix}(g)=\varnothing$}
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 $\sum_g|\operatorname{Fix}(g)|=\sum_x|G_x|=|G|$. 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.
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