= Group of order eight
{title2=$|G|=8$}
= Groups of order eight
{synonym}
Up to <group isomorphism>, the <groups of order eight> are $C_8$, $C_4\times C_2$, $C_2^3$, $D_8$ and $Q_8$, with $D_8$ here denoting the eight-element <dihedral group>. An element of order eight gives the first case. If every nonidentity element has order two the <group> is an <elementary abelian group>. Otherwise, after excluding elements of order eight, a <cyclic subgroup> of order four has index two; an element outside it either commutes with its generator or inverts it. In the latter case its square is either the identity or the central <involution>, giving the <dihedral group> or <quaternion group>.
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