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Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups $G$, i.e., the Steenrod operations on $H^*(BG)$, has ever been done. The case for $G = U(n)$ or torus and $p =2$ is probably easy, but I cannot find reference for more general situations.

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    $\begingroup$ Have you looked at the book by Milnor and Stasheff, on characteristic classes? $\endgroup$ Commented Feb 14 at 20:41
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    $\begingroup$ You’ll find “Wu formula” and “characteristic class” to be useful search keywords. $\endgroup$ Commented Feb 15 at 2:01
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    $\begingroup$ The most complete reference(s) I know of in this regard are the monographs The topology of Lie groups, I and II by Mimura and Toda. $\endgroup$
    – Mark Grant
    Commented Feb 19 at 15:23

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With apologies for reviving this old question: the following paper by Borel and Serre seems to be the canonical reference for Steenrod powers in the cohomology of $BU(n)$, $BSO(n)$ and $BSp(n)$:

Borel, Armand; Serre, Jean-Pierre, Groupes de Lie et puissances réduites de Steenrood, Am. J. Math. 75, 409-448 (1953). ZBL0050.39603.

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    $\begingroup$ Thanks! It sounds reasonable that some "ancient" people wrote about it. $\endgroup$
    – UVIR
    Commented May 22 at 12:38

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