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Understanding Cyclic Groups: Generators of ℤ6, ℤ10, and ℤ9

This document explores the properties of cyclic groups, specifically focusing on ℤ6, ℤ10, and ℤ9. It confirms that ℤ6, ℤ10, and ℤ9 are cyclic groups, discussing the number of generators each possesses. The Klein 4-group's non-cyclic nature is also addressed, alongside proof strategies for demonstrating whether groups are abelian or cyclic. The proof structure is highlighted, including key steps involving the definition of group operations and the selection of elements. This information is valuable for students and enthusiasts of abstract algebra.

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Understanding Cyclic Groups: Generators of ℤ6, ℤ10, and ℤ9

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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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