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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 $Σ_{n}$ . However, if $X$ is a $p$ -complete, homotopy associative, homotopy commutative $H$ -space one can define a homotopy action of ${GL}_{n} (ℤ_{p})$ on $X^{n}$ . In various cases, e.g. if multiplication by $p^{r}$ is null homotopic then we get a homotopy action of $G L_{n} (ℤ / p^{r})$ for some $r$ . After one suspension this allows one to split $X^{n}$ using idempotents of $𝔽_{p} {GL}_{n} (ℤ / p)$ which can be lifted to $𝔽_{p} {GL}_{n} (ℤ / p^{r})$ . In fact all of this is possible if $X$ is an $H$ -space...

p-Nilpotence, classifying space indecomposability, and other properties of almost all finite groups.

Hans-Werner Henn; Stewart Priddy — 1994

Commentarii mathematici Helvetici

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Transfer and complex oriented cohomology rings.

Group extensions and automorphism group rings.

A Cellular Construction of BP and Other Irreducible Spectra.

Homotopy splittings involving G and G/O.

Stable Naturality of Dyer-Lashof Structure Maps.

On the Homology of Non-Connected Monoids and Their Associated Groups

Stable splittings associated with Chevalley groups, I.

Stable splittings associated with Chevalley groups, II.

On certain homotopy actions of general linear groups on iterated products

p-Nilpotence, classifying space indecomposability, and other properties of almost all finite groups.