= Dihedral splitting when the half-rotation order is odd
Let $r$ have order $2m$ with $m$ odd, and let $s^2=1$, $srs=r^{-1}$. The <subgroup> $H=\langle r^2,s\rangle$ is a dihedral group of order $2m$, while $K=\langle r^m\rangle$ is a central <cyclic group> of order two. Since $m$ is odd, $H\cap K=\{1\}$ and $r$ lies in $HK$ by <Bézout's identity>. Thus the original group of order $4m$ is the <direct product of groups> $H\times K$. This statement specifies group orders and avoids the two competing conventions for dihedral notation.
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