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

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

Proof.

By the division algorithm, there exist $q,r\\in{\\mathbb{Z}}$ with $0\\leq r<\\operatorname{exp}~{}G$ such that $|G|=q(\\operatorname{exp}~{}G)+r$ . Let $g\\in G$ . Then $e_{G}=g^{|G|}=g^{q(\\operatorname{exp}~{}G)+r}=g^{q(\\operatorname{exp}~{}G)}g^{%r}=(g^{\\operatorname{exp}~{}G})^{q}g^{r}=(e_{G})^{q}g^{r}=e_{G}g^{r}=g^{r}$ . 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 $|G|$ .∎

单词	ProofThatoperatornameexpGDividesG
释义	proof that $\\operatorname{exp}~{}G$ divides $\|G\|$ The following is a proof that $\\operatorname{exp}~{}G$ divides $\|G\|$ for every finite group $G$ . Proof. By the division algorithm, there exist $q,r\\in{\\mathbb{Z}}$ with $0\\leq r<\\operatorname{exp}~{}G$ such that $\|G\|=q(\\operatorname{exp}~{}G)+r$ . Let $g\\in G$ . Then $e_{G}=g^{\|G\|}=g^{q(\\operatorname{exp}~{}G)+r}=g^{q(\\operatorname{exp}~{}G)}g^{%r}=(g^{\\operatorname{exp}~{}G})^{q}g^{r}=(e_{G})^{q}g^{r}=e_{G}g^{r}=g^{r}$ . 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 $\|G\|$ .∎
随便看	proof of Ingham Inequality ProofOfInjectiveImagesOfBaireSpace proof of injective images of Baire space ProofOfIntegralTest proof of integral test ProofOfIntermediateValueTheorem proof of intermediate value theorem ProofOfInvarianceOfDimension proof of invariance of dimension ProofOfInverseFunctionTheorem proof of inverse function theorem ProofOfInverseFunctionTheoremtopologicalSpaces proof of inverse function theorem (topological spaces) ProofOfInverseOfMatrixWithSmallrankAdjustment proof of inverse of matrix with small-rank adjustment ProofOfInvertibleIdealsAreProjective proof of invertible ideals are projective ProofOfJacobisIdentityForvarthetaFunctions proof of Jacobi’s identity for ϑ functions ProofOfJensensInequality proof of Jensen’s inequality ProofOfJordanCanonicalFormTheorem proof of Jordan canonical form theorem ProofOfJordansInequality proof of Jordan’s Inequality

proof that exp⁡G divides |G|

Proof.

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