I am trying to learn a bit about (geometrical Euclidean) group theory and am now wondering "Wat is so special about the Special Orthogonal group?"
Or what is the difference between the groups $O(2)$ and $SO(2)?$
Or name some transformations belong to $O(2)$ but not to $SO(2)$
the same for the Linear Group and the Special Linear group $SL(2)$.
$\endgroup$ 32 Answers
$\begingroup$Here's an elaboration of my comment.
Everything works for $SO(n)$ and $O(n)$ so I'll write it in that generality. The subgroup $SO(n)$ is the kernel of the determinant homomorphism $det : O(n) \to \{\pm 1\}$. Since $det$ is surjective and continuous, $O(n)$ is not connected.
But the group $SO(n)$ is path connected: if the columns of matrix $M \in SO(n)$ are $v_1,v_2,\ldots,v_n$, and so $v_1,…,v_n$ form a positively oriented orthonormal basis, one first rotates the basis continuously so that $v_1$ lines up with $e_1$; then holding $e_1$ fixed one rotates continuously so that $v_2$ lines up with $e_2$; etc. At the end of the process, $v_1,\ldots,v_{n-1}$ have been rotated continuously to line up with $e_1,\ldots,e_{n-1}$, and so $v_n$ must line up with $\pm e_n$, but it must be $+e_n$ since the determinant is positive. These "continuous rotations" concatenate together to give the desired continuous path in $SO(n)$ from $M$ to the identity matrix.
Since the coset $O(n)-SO(n)$ is homeomorphic to $O(n)$, it follows that $O(n)$ has exactly two components: the subgroup $SO(n)$; and the coset $O(n)-SO(n)$.
$\endgroup$ $\begingroup$In this context it just means each element has determinant 1.
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