Let $G$ be a compact group (may be non-Lie group). . Let $B_{G}$ denote the
classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$,
then I think that $H^{\ast }\left( B_{G};%
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\mathbb{Q}
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\right) $ is not trivial, where $H^{\ast }$ is the cohomology algebra. I don't
know how exactly to prove this.

I just know this: If $G$ is a torus and $H$ is a subtorus of $G$, then $%
H^{\ast }\left( B_{G};%
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\mathbb{Q}
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\right) \longrightarrow H^{\ast }\left( B_{H};%
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\mathbb{Q}
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\right) $ is surjective.