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Homework6

# Homework6 - Homework 6 Solutions 1 Table of Elements and...

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Homework 6 Solutions 1. Table of Elements and their Orders: element 1 7 43 49 51 57 93 99 101 107 143 149 151 157 193 199 order 1 4 4 2 2 4 4 2 2 4 4 2 2 4 4 2 Since | G | = 16 and G only has elements of 4 or 2 (or 1) we know either G = Z 4 × Z 4 or G = Z 4 × Z 2 × Z 2 . Z 4 × Z 2 × Z 2 has 8 elements of order 4 yet Z 4 × Z 4 has 12 elements of order 4. Thus we can conclude G = Z 4 × Z 2 × Z 2 . 2. (a) If G is an abelian group of order 9 then G is isomorphic to either Z 9 or Z 3 × Z 3 . The former has 6 elements of order 9, 2 elements of order 3 and an identity. The latter has 8 elements of order 3 and an identity. Thus if you calculate the order of 3 elements (besides the identity) you can deﬁnitively determine the isomorphism class of G . (b) If G is an abelian group of order 18 then G is isomrphic to either Z 2 × Z 3 × Z 3 or Z 2 × Z 9 . The former has 8 elements of order 3, 1 element of order 2, 8 elements of order 6 and an identity. The latter has 2 elements of order 3, 6 elements of order 9, 1 element of order
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