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Unformatted text preview: Algebra , Assignment 8 Solutions 1. Three problems about manipulating products of cyclic groups. (a) Write Z 2 Z 2 Z 2 Z 9 Z 5 Z 25 as the product of cyclic groups Z a i , 1 i s (you find the s ) with a i dividing a i +1 for all 1 i < s . Solution: We line up each power separately right justified: Z 2 Z 2 Z 2-- Z 9- Z 5 Z 25 Multiplying down the columns (note the numbers are relatively prime in the column so this can be done) gives the answer Z 2 Z 10 Z 450 . (b) Write Z 4 Z 40 Z 200 Z 1400 as the product of cyclic groups of prime power order. (Prime power includes primes themselves.) Solution: Break each up into its prime powers giving Z 4 Z 8 Z 5 Z 8 Z 25 Z 8 Z 25 Z 7 (c) Write Z 5 Z 6 Z 7 in both of the above forms. Solution: For the second we have Z 5 Z 2 Z 3 Z 7 . For the first each prime power appears only once so we have Z 210 . 2. Let G = Z 5 Z 25 Z 125 . Find the order of (3 , 10 , 12). Give a good description and a precise count on those ( a, b, c ) G order 25....
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This note was uploaded on 04/01/2011 for the course MATH 101 taught by Professor Josephs during the Fall '08 term at NYU.
- Fall '08