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proof that $\operatorname{exp}~G$ divides $| G |$


The following is a proof that $\operatorname{exp}~G$ divides $|G|$ for every finite group $G$.

By the division algorithm, there exist $q,r \in {\mathbb Z}$ with $0 \le r<\operatorname{exp}~G$ such that $|G|=q(\operatorname{exp}~G)+r$.  Let $g \in G$.  Then $e_G=g^{|G|}=g^{q(\operatorname{exp}~G)+r}=g^{q(\operatorname{exp}~G)}g^r=(g^{\operatorname{exp}~G})^qg^r=(e_G)^qg^r=e_Gg^r=g^r$.  Thus, for every $g \in G$, $g^r=e_G$.  By the definition of exponent, $r$ cannot be positive.  Thus, $r=0$.  It follows that $\operatorname{exp}~G$ divides $|G|$.