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Find all subgroups of the Klein 4- Group. How many are there? 1 2 3 4 5 6 7 8 9 10.

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Presentation on theme: "Find all subgroups of the Klein 4- Group. How many are there? 1 2 3 4 5 6 7 8 9 10."— Presentation transcript:

1 Find all subgroups of the Klein 4- Group. How many are there? 1 2 3 4 5 6 7 8 9 10

2 Find all subgroups of Z 4. How many are there? 1 2 3 4 5 6 7 8 9 10

3 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.

4 What is the next line in this proof? (a) Then G is a subgroup of H. (b) Then G contains inverses. (c) Let a, b be any two elements in G. (d) Let H be any subgroup in G.

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

6 What is the second to last line in this proof? (a) Then G is cyclic. (b) Then G has finite order. (c) Then H = for some ? in G. (d) Then H has finite order.

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