Download Example. Let G be the positive real numbers with multiplication and

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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.