Show that if $G$ and $H$ are groups whose orders are coprime, then $BG$ indexed limits commute with $BH$-indexed colimits in $Set$.

Question

How do I show that if $G$ and $H$ are groups whose orders are coprime, then $BG$-indexed limits commute with $BH$-indexed colimits in Set? I would just like hints.

Answer 1

Before giving some hints, one must first unwind all the definitions to understand the situation :

We are interested in a functor $X: BG\times BH\to \mathbf{Set}$, that is, a set $X$ together with a $G$-action and an $H$-action that commute.

Taking colimit along $BH$ yields the orbit space $X/H$ and then limit along $BG$ is taking its invariants (from the induced action) : $(X/H)^G$.

Taking the limit first amounts to taking the invariants $X^G$ on which $H$ acts, and then taking the colimit along $BH$ is taking the orbit space of that : $X^G/H$.

There is a natural map $X^G/H\to (X/H)^G$ which takes the class of an invariant to its class mod $H$ (which is $G$-invariant)

The statement is that this is a bijection.

Here are some hints to prove this :

-surjectivity : take a $G$-invariant $H$-orbit. It means the $G$-action on $X$ restricts to that orbit. But what is the cardinality of that orbit ? Can it have no fixed points in it ?

-injectivity : This is independent of the hypothesis on $|H|, |G|$, you should try to prove it.