Find all subgroups of the Klein 4-Group. How many are there?

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 Z4 . How many are there?
1 2 3 4 5 6 7 8 9 10
3

• 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.

4

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

5

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

6

• What is the second to last line in this proof?
• Then G is cyclic.
• Then G has finite order.
• Then H lt?gt for some ? in G.
• Then H has finite order.