Cyclic groups as normal sobgroups of SO(3) I have to argue why none of the cyclic groups $C_n$ is a normal subgroup of $SO(3)$. Nevertheless, I haven't found an argument for that. First I used the definition of normal subgroup: If some $C_n$ were normal subgroup of $SO(3)$, then would be fulfilled that $GcG^{-1}\in C_n$, $\forall G\in SO(3)$ and $\forall c\in C_n$.
I tried to begin with this; Let $C_n$ a normal subgroup of $SO(3)$. If $c\in C_n$ and $G\in SO(3)$, then $GcG^{-1}=\tilde{c}$ for some $\tilde{c}\in C_n$. But if $d$ is the generator of $C_n$, then $c=d^{i}=d \cdot d\cdots d$ (i-times) and $\tilde{c}=d^{j}$, with $i,j\in\{0,1,\dots,n-1\}$. Then $Gd^{i}G^{-1}=d^{j}$. But after this, I cannot see how the features of $SO(3)$ could help me. I mean, I know that $SO(3)$ are square orthogonal matrices with determinant equal to 1, but this seems not to be useful in this case, well not as far as I see.
 A: Every non-identity element of $SO(3)$ has exactly one eigenvalue equal to 1. Geometrically (viewing elements of $SO(3)$ as linear transformations of $\mathbb{R}^3$) speaking this means that every such element fixes exactly one line through the origin.
When the element generates a cyclic group, this whole group fixes that line. Since being orthogonal means 'preserving the inner product' which in more down to earth geometric terms means 'preserving distances and angles' we see that elements of the cyclic subgroup of $SO(3)$ are rigid motions preserving a line, hence rotations over some angle around that line.
Now think about the line fixed by your group as running through the North and Southpole of some sphere and look again at your $GcG^{-1}$ for the special case that $G$ is a rotation over 90 degrees around an axis running 'front to back' through the equator (i.e. an axis perpendicular to the axis fixed by the elements of the cyclic group). Try to understand what the rotation $GcG^{-1}$ does to an easy to visualize point on the sphere like the North pole, or some other point, and conclude that $GcG^{-1}$ is not in the cyclic group.
