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Unformatted text preview: 148 3. PRODUCTS OF GROUPS symmetries of bananas: . ; /. ; / D . ; / . Thus the symmetry group is the direct product A B S 5 S 4 . Example 3.1.3. Let a and b be relatively prime natural numbers, each greater than or equal to 2. We showed in the proof of the Chinese remain der theorem near the end of Section 1.11 that the map OEx ab 7! .OEx a ;OEx b / from Z ab to Z a Z b is well defined and bijective. It is straightforward to check that this is an isomorphism of groups. Thus Z ab Z a Z b : Example 3.1.4. Let a and b be relatively prime natural numbers, each greater than or equal to 2. The bijective map W OEx ab 7! .OEx a ;OEx b / from Z ab to Z a Z b is in fact a ring isomorphism; this is easy to check, and was given as Exercise 1.11.10 . It follows that maps the set .ab/ of elements with multiplicative inverse in Z ab bijectively onto the set el ements with multiplicative inverse in Z a Z b . Since respects mul tiplication, in particular it respects multiplication of invertible elements,...
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 Fall '08
 EVERAGE
 Algebra, Natural Numbers

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