Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)

The set $\text{Fix}_H(X)$ of $H$-fixed points carries a $G$-action, and we can then take the set $\text{Fix}_H(X)/G$ of orbits. But also, the set $X/G$ of $G$-orbits carries an $H$-action, and we can take the set $\text{Fix}_H(X/G)$ of fixed points. There is a canonical map $$ \lambda\colon \text{Fix}_H(X)/G \to \text{Fix}_H(X/G), $$ which is easily seen to be injective.

It's a fact that if $G$ and $H$ are finite groups with coprime orders, this map $\lambda$ is bijective. So then, $$ \text{Fix}_H(X)/G \cong \text{Fix}_H(X/G). $$ (A bit more generally, this is true whenever $G$ and $H$ are possibly-infinite groups with finite coprime exponents.) The proof isn't hard.

I only discovered this yesterday, but I guess it's well-known — maybe as a special case of something more general. Can anyone give me a reference?

**Update** Perhaps I should have explained the context. In category theory, limits commute with limits and colimits commute with colimits, but limits do not usually commute with colimits. There are various theorems giving restricted conditions under which limits *do* commute with colimits. This result about group actions, which Peter Johnstone and I came upon, gives a new (?) such theorem. That's why I'm after a reference. For more explanation, see this $n$-Category Café post.

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