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Let $K$ be a normal subgroup of order 2 in group $G$, show that $K$ lies in the centre of $G$. Describe a surjective homomorphism of the orthogonal group $\mathrm{O}(3)$ onto $C_2$ and another onto the special orthogonal group $\mathrm{SO}(3)$.

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Since $K$ is a normal subgroup of order $2$, there is only one nonidentity element, say $a\in K$. Then for all $g\in G$, we have that $gag^{-1}=e$ or $gag^{-1}=a$. If the latter occurs, then $K\subseteq Z(G)$ and we're done since $(gag^{-1})g=ga=ag$. Assume that $gag^{-1}=e$. Then $ga=g$, thus multiplying on the left by $g^{-1}$, we get $a=e$, a contradiction. Thus, $K\subseteq Z(G)$ as desired.

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Hint: Prove that $[x,y]\in K$ for every $y\in G$, where $x$ is the generator of $K$.

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$[x,y]=x^{-1}y^{-1}xy$. +1 – Babak S. Dec 30 '12 at 9:37

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