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

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Presentation transcript:

Is ℤ 6 a cyclic group? (a) Yes (b) No

How many generators are there of ℤ 6 ?

How many generators are there of ℤ 10 ?

How many generators are there of ℤ 9 ?

Is the Klein 4-Group a cyclic group? (a) Yes (b) No

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

What is the next line in this proof? (a) Then G is abelian. (b) Then G contains inverses. (c) Then a * b = b * a for all a, b in G. (d) Then G = for some x in G.

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

What is the last line in this proof? (a) Thus G is abelian. (b) Thus G contains inverses. (c) Therefore G is cyclic. (d) Then G has primary order.

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