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Material Type: Quiz; Class: Group Theory; Subject: Mathematics; University: Northeastern University; Term: Fall 2010;
Typology: Quizzes
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Prof. Alexandru Suciu
MATH 3175 Group Theory Fall 2010
(b) Prove that G is not cyclic.
MATH 3175 Quiz 5 Fall 2010
MATH 3175 Solutions to Quiz 5 Fall 2010
#{elements of order 7 in Z 70 } = ฯ(7) = 6 #{elements of order 7 in Z 490 } = ฯ(7) = 6 #{of elements of order 7 in Z 70 โ Z 490 } = ฯ(7) ร 1 + ฯ(7) ร ฯ(7) + 1 ร ฯ(7) = 6 + 6 ร 6 + 6 = 48
WARNING: Since the problem asks you to write down the groups as direct prod- ucts of cyclic groups of prime power order, you cannot write for example Z 72 instead of Z 23 ร Z 32 , even though the two groups are isomorphic (because 2^3 is prime to 3^2 ).
The order of G has prime factorization 108 = 2^2 ร 33. The abelian groups of order 108 (up to isomorphism) are:
Z 22 โ Z 32 โ Z 3 (This group satisfies the conditions.) Z 22 โ Z 3 โ Z 3 โ Z 3 (This group has 26 elements of order 3 and 1 element of order 2.) Z 2 โ Z 2 โ Z 33 Z 2 โ Z 2 โ Z 32 โ Z 3 Z 2 โ Z 2 โ Z 3 โ Z 3 โ Z 3
