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May 3, 2019 at 19:22 comment added John Rognes @Dr.Martens Arunas Liulevicius proved (in his 1960 PhD thesis, published in 1962 as an AMS Memoir), that $P^{p^i}$ factors as a secondary cohomology operation for $i\ge1$ when $p$ is odd. These factorizations lead to nonzero $d_2$-differentials in the Adams spectral sequence for the sphere spectrum, so reading e.g. the introduction in Ravenel's "Complex cobordism and stable homotopy groups of spheres" will put these results in a better context.
May 3, 2019 at 8:23 comment added Dr.Martens What is the reference to the same question $mod p$ about quotients $A_p/A_p(\beta,P^1,P^p,\ldots,P^{p^k})$ and $A_p/A_p(P^1,P^p,\ldots,P^{p^k})$?
May 3, 2019 at 7:05 history answered John Rognes CC BY-SA 4.0