= Commutation of fixed points and orbit quotients for coprime groups
{title2=$A^G/H\cong(A/H)^G$}
For commuting actions of finite <groups> $G,H$ with <coprime> orders, the canonical map $A^G/H\to(A/H)^G$ is a bijection. It is always injective. On any $G$-stable $H$-orbit, all $G$-orbits have the same size, dividing both $|G|$ and the $H$-orbit size, hence dividing $|H|$. Coprimality forces that size to be one, proving surjectivity. Thus <categorical limits> indexed by $G$ commute with <colimits> indexed by $H$ in the <Category of sets>.
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