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Is ℤ 6 a cyclic group? Yes No

Is ℤ 6 a cyclic group? Yes No. How many generators are there of ℤ 6 ? 1 2 3 4 5 6 7 8 9 10. How many generators are there of ℤ 10 ? 1 2 3 4 5 6 7 8 9 10. How many generators are there of ℤ 9 ? 1 2 3 4 5 6 7 8 9 10.

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Is ℤ 6 a cyclic group? Yes No

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  1. Is ℤ6 a cyclic group? • Yes • No

  2. How many generators are there of ℤ6 ? 1 2 3 4 5 6 7 8 9 10

  3. How many generators are there of ℤ10 ? 1 2 3 4 5 6 7 8 9 10

  4. How many generators are there of ℤ9 ? 1 2 3 4 5 6 7 8 9 10

  5. Is the Klein 4-Group a cyclic group? • Yes • No

  6. What is the first line in this proof? • Assume G is an abelian group. • Assume G is a cyclic group. • Assume a * b = b * a.

  7. What is the next line in this proof? • Then G is abelian. • Then G contains inverses. • Then a * b = b * a for all a, b in G. • Then G = <x> for some x in G.

  8. What is the next line in this proof? • Choose any two elements of G. • Then G has finite order. • Then a * b = b * a for all a, b in G. • Choose any element x and its inverse.

  9. What is the last line in this proof? • Thus G is abelian. • Thus G contains inverses. • Therefore G is cyclic. • Then G has primary order.

  10. What is the second to last line in this proof? • Then G is cyclic. • Then G has finite order. • Then a * b = b * a for all a, b in G. • Choose any element x and its inverse.

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