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The $n$-fold product $X^n$ of an arbitrary space usually supports only the obvious permutation action of the symmetric group ${\Sigma }_n$. However, if $X$ is a $p$-complete, homotopy associative, homotopy commutative $H$-space one can define a homotopy action of ${\mathrm{GL}}_n\left({\mathbb{Z}}_p\right)$ on $X^n$. In various cases, e.g. if multiplication by $p^r$ is null homotopic then we get a homotopy action of $\mathrm{G}{L}_n(\mathbb{Z}/p^r)$ for some $r$. After one suspension this allows one to split $X^n$ using idempotents of ${𝔽}_p{\mathrm{GL}}_n(\mathbb{Z}/p)$ which can be lifted to ${𝔽}_p{\mathrm{GL}}_n(\mathbb{Z}/p^r)$. In fact all of this is possible if $X$ is an $H$-space...