= Normal subgroups of coprime order commute
If finite <normal subgroups> $H,K$ of a <group> have coprime orders, their intersection is trivial by <Lagrange's theorem>. For $h\in H$ and $k\in K$, the <commutator> $hkh^{-1}k^{-1}$ belongs to both subgroups: write it as $h(kh^{-1}k^{-1})$ to use normality of $H$, and as $(hkh^{-1})k^{-1}$ to use normality of $K$. Hence it is the identity and $hk=kh$. The product subgroup is therefore isomorphic to the <direct product of groups> $H\times K$.
Back to article page
