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72 6. ISOMORPHISMS Example. Let G be the positive real numbers with multiplication and G be the group of real numbers with addition. Show G ⇡ G. Proof. Define : G ! G by (x) = ln(x). [Show 1–1.] Suppose (x) = (y). Then ln(x) = ln(y) =) eln(x) = eln(y) =) x = y, so is 1–1. [Show onto] Now suppose x 2 G. ex > 0 and (ex) = ln ex = x, so is onto. [Show operation preservation.] Finally, for all x, y 2 G, (xy) = ln(xy) = ln x + ln y = (x) + (y), so G ⇡ G. [Question: is the only isomorphism?] ⇤ Example. Any infinite cyclic group is isomorphic to Z with addition. Given hai with |a| = 1, define (ak ) = k. The map is clearly onto. If (an) = (am), n = m =) an = am, so is 1–1. Also, (anam) = (an+m) = n + m = (n) + (m), so hai ⇡ Z. Consider h2i under addition, the cyclic group of even integers. The h2i ⇡ Z = h1i with : h2i ! Z defined by (2n) = n.