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

The following is a proof that, for every group $G$ that has an exponent and for every $g\\in G$ , $|g|$ divides $\\operatorname{exp}~{}G$ .

Proof.

By the division algorithm, there exist $q,r\\in{\\mathbb{Z}}$ with $0\\leq r<|g|$ such that $\\operatorname{exp}~{}G=q|g|+r$ . Since $e_{G}=g^{\\operatorname{exp}~{}G}=g^{q|g|+r}=(g^{|g|})^{q}g^{r}=(e_{G})^{q}g^{r%}=e_{G}g^{r}=g^{r}$ , by definition of the order of an element, $r$ cannot be positive. Thus, $r=0$ . It follows that $|g|$ divides $\\operatorname{exp}~{}G$ .∎

单词	ProofThatgDividesoperatornameexpG
释义	proof that $\|g\|$ divides $\\operatorname{exp}~{}G$ The following is a proof that, for every group $G$ that has an exponent and for every $g\\in G$ , $\|g\|$ divides $\\operatorname{exp}~{}G$ . Proof. By the division algorithm, there exist $q,r\\in{\\mathbb{Z}}$ with $0\\leq r<\|g\|$ such that $\\operatorname{exp}~{}G=q\|g\|+r$ . Since $e_{G}=g^{\\operatorname{exp}~{}G}=g^{q\|g\|+r}=(g^{\|g\|})^{q}g^{r}=(e_{G})^{q}g^{r%}=e_{G}g^{r}=g^{r}$ , by definition of the order of an element, $r$ cannot be positive. Thus, $r=0$ . It follows that $\|g\|$ divides $\\operatorname{exp}~{}G$ .∎
随便看	KateFenchel KathleenOllerenshaw Kathleen Ollerenshaw KatoRellichTheorem Kato-Rellich theorem KautzGraph Kautz graph KazimierzZarankiewicz Kazimierz Zarankiewicz kconnectedGraph k-connected graph KdistanceSet K-distance set Keepflipchange keep-flip-change KeithNumber Keith number KempeChain Kempe chain KempnerSeries Kempner series KenosymplirosticNumbers Kenosymplirostic numbers Kernel kernel

proof that |g| divides exp⁡G

Proof.

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