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# sol8 - Algebra Assignment 8 Solutions 1 Three problems...

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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. Solution: i (3 , 10 , 12) = (0 , 0 , 0) requires 5 | i , 5 | i and 125 | i so it need 125 | i so the order is 125. (The order must be a power of 5 as o

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sol8 - Algebra Assignment 8 Solutions 1 Three problems...

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