**Theorem:** For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally isomorphic to the character group $%
\widehat{G}$. If $G$ is connected, then the entire cohomology algebra $%
H^{\ast }\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ is the symmetric $%
%TCIMACRO{\U{2124} }%
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\mathbb{Z}
%EndExpansion
$-algebra $P\left( \widehat{G}\right) $ generated by this component of
homogeneous degree $2$.

**Observation:** If $G$ is a compact connected abelian group and $N$ any closed
normal subgroup, then the quotient morphism $q:G\longrightarrow G/N$ induces
an injection $\widehat{q}:\widehat{G/N}\cong N^{\bot }\longrightarrow 
\widehat{G}$ by Pontryagin Duality with annihilator $N^{\bot }$ of $N$ in $%
\widehat{G}$.

Someone says that the symmetric algebra functor $P$ preserves inclusions, we
have $P\left( N^{\bot }\right) \subset P\left( \widehat{G}\right) $, and so $%
H^{\ast }\left( q;%
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\mathbb{Z}
%EndExpansion
\right) :H^{\ast }\left( B_{G/N};%
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%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) \longrightarrow H^{\ast }\left( B_{G};%
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\mathbb{Z}
%EndExpansion
\right) $ is injective.

Another person says that is not true that the symmetric algebra functor $P$
preserves inclusions.

Which one of these is correct? That is  does the symmetric algebra functor $P
$ preserve inclusions?

**Edit**

[![enter image description here][1]][1]

I found this note in an article:
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/tL7Uq.png
  [2]: https://i.sstatic.net/YbrK8.png