alg-01-6 - 1.6. CYCLIC SUBGROUPS 19 1.6 Cyclic Subgroups...

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1.6. CYCLIC SUBGROUPS 19 1.6 Cyclic Subgroups Recall: cyclic subgroup, cyclic group, generator. Def 1.68. Let G be a group and a G . If the cyclic subgroup h a i is ﬁnite, then the order of a is |h a i| . Otherwise, a is of inﬁnite order . 1.6.1 Elementary Properties Thm 1.69. Every cyclic group is abelian. Thm 1.70. If m Z + and n Z , then there exist unique q,r Z such that n = mq + r and 0 r m. In fact, q = b n m c and r = n - mq . Here b x c denotes the maximal integer no more than x . Ex 1.71 (Ex 6.4, Ex 6.5, p60). 1. Find the quotient q and the remainder r when n = 38 is divided by m = 7. 2. Find the quotient q and the remainder r when n = - 38 is divided by m = 7. Thm 1.72 (Important). A subgroup of a cyclic group is cyclic. Proof. (refer to the book) Ex 1.73. The subgroups of h Z , + i are precisely h n Z , + i for n Z . Def 1.74. Let r,s Z . The greatest common divisor (gcd) of r and s is the largest positive integer d that divides both r and s . Written as

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This note was uploaded on 10/12/2010 for the course MATH 5310 taught by Professor Staff during the Spring '08 term at Auburn University.

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alg-01-6 - 1.6. CYCLIC SUBGROUPS 19 1.6 Cyclic Subgroups...

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