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# Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varp... ## Question ###### Let$G=\mathbb{Q}^{x}$be the multiplicative group of nonzero rational numbers. If$\alpha=p / q \in G$, where$p$and$q$are relatively prime integers, let$\varphi: G \rightarrow G$be the map which interchanges the primes 2 and 3 in the prime power factorizations of$p$and$q$(so, for example,$\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and$\varphi$is the identity on all rational numbers with n Let$G=\mathbb{Q}^{x}$be the multiplicative group of nonzero rational numbers. If$\alpha=p / q \in G$, where$p$and$q$are relatively prime integers, let$\varphi: G \rightarrow G$be the map which interchanges the primes 2 and 3 in the prime power factorizations of$p$and$q$(so, for example,$\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and$\varphi$is the identity on all rational numbers with numerators and denominators relatively prime to 2 and to 3 ). (a) Prove that$\varphi$is a group isomorphism. (b) Prove that there are infinitely many isomorphisms of the group$G$to itself. (c) Prove that none of the isomorphisms above can be extended to an isomorphism of the ring$\mathbb{Q}$to itself. In fact prove that the identity map is the only ring isomorphism of$\mathbb{Q}$. ## Answers #### Similar Solved Questions 5 answers ##### Suppose a student failed to account for the partial pressure of water vapor in his molar mass determination. 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An investment club has set goal of earning 1590 on the mone invests in stocks The members considering purchasing three possible stocks vith their cost per share (in dollars) their projected grovth per share dollars) summarized in the folloving table (Let * computershares utility shares and retail s...
##### Find the inverse of the Fourier transform of F(w) 21(6 + 5iw[convolution]
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##### 1 the M standard Obsanyed t0 L W Unit symboi 1 Oharmation 0l compound 8 necessany formula (You may Jssume 1 at 25 It to Dennd e II constant-pre %urc 1| (ormeter containing 1 1 1 3 1
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##### Below is the comic that E near the end of the lecture slides on inductive proofs: It is silly; but some important points are made. Explain in your own words why this inductive proof is wrong: Identify all the errors thatyou can:Base Case any 0ne dinosaur the same colour as itself:All cnosaus are the same colour!FALSE MNDUCTIVE PROOF COMICS0f couise,Assume that any group ofndnosaurs the same cclour. Consider Orouc n+ldnosauis; The frst n (dno ton) are the same cciouUr.And the LAST n (dino ton+l)
Below is the comic that E near the end of the lecture slides on inductive proofs: It is silly; but some important points are made. Explain in your own words why this inductive proof is wrong: Identify all the errors thatyou can: Base Case any 0ne dinosaur the same colour as itself: All cnosaus are t...
##### QUESTION 12. If the electric potential at the point P for this electric dipole is 4Sx10V. What is the dipole moment?PZmm
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(2r _ +vY = '(2r' 9y + (QW/2yis a polynomial identity Determine if the equation...