= Solution
Let $a,b$ be the generating <involutions>. If $a\ne b$, set $x=ab$ and $y=b$. Then $x\ne e$, so its <order of a group element> is at least two or infinite, and
$$
yxy^{-1}=b(ab)b=ba=(ab)^{-1}=x^{-1},\qquad a=xy.
$$
Therefore $x,y$ generate the <group> and satisfy the required dihedral inversion relation. If $a=b$, the <group> is $C_2$ and $x=y=a$ gives its permitted degenerate description. This proves the <two-involution characterization of a dihedral group> without excluding either $C_2$ or the <infinite dihedral group>.
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