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Question: How to calculate the cohomology ring(with $\mathbb{Z}/2$ coefficient) structure of $U(n),$ the unitary group without using Serre spectral sequence. I am interested in $U(3).$

Attemt: Since $(U(3), U(2))$ is $4$-connected. Therefore the inclusion map $i : U(2) \to U(3)$ is an isomorphism in cohomology upto dimension $3.$ Again since $U(2) \cong S^1 \times S^3$, therefore by Kunneth formula $H^*(U(2), \mathbb{Z}/2) \cong \frac{\mathbb{Z/2}[x_1, x_3]}{(x_1^2, \; x_3^2)}.$

Please suggest me some way for computation of the ring structure of $H^*(U(3)).$

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