Is Z6 a cyclic group

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 ltxgt 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.