= Two-involution characterization of a dihedral group
{title2=$G=\langle a,b\rangle,\quad a^2=b^2=e$}
A <group> generated by two <involutions> is dihedral in the extended convention allowing $C_2$ and the <infinite dihedral group>. If $a\ne b$, put $x=ab$, $y=b$; then $yxy^{-1}=x^{-1}$. Conversely, from the inversion relation, $y$ and $xy$ square to the identity and generate. If $xy=e$, the group is $C_2$ and the same nonidentity element is used twice. Exact order two, rather than merely a square equal to the identity, requires this exceptional case. This convention extends the usual finite <dihedral group> definition.
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