Solution
= Solution
{c}
Among the $|G|+1$ powers $e,g,g^2,\ldots,g^{|G|}$ in the finite <group> $G$, the <pigeonhole principle> forces two to be equal: $g^i=g^j$ with $0\leq i<j\leq|G|$. Multiply by $g^{-i}$ to obtain
$$
\boxed{g^{j-i}=e,\qquad j-i>0.}
$$
This proves the existence of a positive power equal to the <identity element>, without presupposing the <order of a group element>.