College Algebra Exam Review 138

College Algebra Exam Review 138 - 148 3. PRODUCTS OF GROUPS...

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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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