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 $\mathrm{SO}(3)$.
Let $K$ be a normal subgroup of order 2 in group $G$, show that $K$ lies in the centre of $G$
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group-theory