= Classification of groups of order ten
{title2=$C_{10},\ D_{10}$}
A <group> of order ten is either a <cyclic group> or the <dihedral group> of the pentagon. <Lagrange's theorem> and the fact that a group of exponent two has power-of-two order force an element of order five or ten. In the latter case the group is cyclic. In the former case its order-five <cyclic subgroup> is normal by index two; an outside involution acts on it either trivially or by inversion. The trivial action gives $C_5\times C_2\cong C_{10}$, and inversion gives $D_{10}$.
Back to article page
