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College Algebra Exam Review 91

College Algebra Exam Review 91 - 101 2.2 SUBGROUPS AND...

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2.2. SUBGROUPS AND CYCLIC GROUPS 101 Z 12 @ @ @ @ @ h OE3Ł i h OE2Ł i @ @ @ @ @ @ @ @ @ @ h OE6Ł i h OE4Ł i @ @ @ @ @ f 0 g Figure 2.2.2. Lattice of subgroups of Z 12 . integers s such that s is congruent modulo n to an integer multiple of b , or, equivalently, the smallest of positive integers s such that OEsŁ 2 h OEbŁ i . By the proof of Proposition 2.2.24 , h OEdŁ i D h OEbŁ i . The order of OEbŁ is the order of h OEbŁ i , which is n=d , by Proposition 2.2.24 (b). Part (c) is left as an exercise. n Example 2.2.29. Find all generators of Z 12 . Find all OEbŁ 2 Z 12 such that h OEbŁ i D h OE3Ł i , the unique subgroup of order 4. The generators of Z 12 are exactly those OEaŁ such that 1 a 11 and a is relatively prime to 12. Thus the generators are OE1Ł; OE5Ł; OE7Ł; OE11Ł
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