Improper Rotations in Even Dimensions
In odd dimensions, we can represent any improper "rotation" G as $-\mathbb{1}\cdot R$ where $R\in SO(d)$. In even dimensions, $-\mathbb{1} \in SO(d)$ and we cant do this. Is there a way of writing an improper rotation in terms of proper rotations in even dimensions? Also, if we have improper rotation G (det = -1) in even dimensions, is there anything general that we can conclude about the order of the cyclic subgroup that it generates? For example, in even dimensions the order of the cyclic group generated by G must be even.
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$\begingroup$Instead take the identity matrix $I$ and change the (1,1) entry to -1 to give the matrix $J$. Certainly has $\det(J)=-1$ and also $JJ^t=I$, so it is orthogonal.
The kernel of the homomorphism $O(n)\rightarrow \{+1,-1\}$ is $SO(n)$ so $SO(n)$ has index 2 in $O(n)$. Thus $O(n)$ is a disjoint union of two left cosets, $O(n)=SO(n) \cup J\cdot SO(n)$.
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