Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 6D / c / Solution

{title2=$O(2m+1)\cong SO(2m+1)\times C_2$}

In odd dimension, the map $A\mapsto(\det A)A$ is a surjective <group homomorphism> from the <orthogonal group> to the <special orthogonal group>, with <kernel of a group homomorphism> $\{I,-I\}$. Scalar matrices commute, and $\det[(\det A)A]=(\det A)^{2m+2}=1$. Together with the <determinant> sign, it identifies $O(2m+1)$ with the <direct product of groups> $SO(2m+1)\times C_2$. In even dimension, scalar multiplication by $\det A$ does not repair the determinant, so this construction does not give the same splitting.

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 Incoming links: Odd-dimensional orthogonal determinant splitting