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In this paper we use Hopf algebra and generating function methods to determine the group of all cohomology morphisms from      H ∗  ( B U ; Z / p ) {H^ * }\left ( {BU;Z/p} \right )       to      H ∗  ( B Z / p ; Z / p ) {H^ * }\left ( {BZ/p;Z/p} \right )       that preserve the Steenrod operations, where      p p       is an odd prime. The group      [ B Z / p , B U ] \left [ {BZ/p,BU} \right ]       of homotopy classes of maps from      B Z / p BZ/p       to BU, which can be calculated directly, is seen to be naturally isomorphic to the group of cohomology morphisms. For      B Z / 2 BZ/2       and BO with coefficients in      Z / 2 Z/2       there are precisely similar results.

Cohomology morphisms from 𝐻*(𝐵𝑈;𝑍/𝑝) to 𝐻*(𝐵𝑍/𝑝;𝑍/𝑝)

Cohomology morphisms from 𝐻(𝐵𝑈;𝑍/𝑝) to 𝐻(𝐵𝑍/𝑝;𝑍/𝑝)