How many different non-isomorphic Abelian groups of order 4 are there
2007
How many different non-isomorphic Abelian groups of order 4 are there
- A.
2
- B.
3
- C.
4
- D.
5
Attempted by 99 students.
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Correct answer: A
Answer: There are 2 non-isomorphic abelian groups of order 4.
Key idea: Use the classification of finite abelian groups. Since 4 = 2^2, consider partitions of the exponent 2.
Partition 2 as 2: this gives the cyclic group of order 4, isomorphic to Z4.
Partition 2 as 1 + 1: this gives the direct product Z2 × Z2 (the Klein four-group).
Conclusion: Therefore there are exactly two non-isomorphic abelian groups of order 4: Z4 and Z2 × Z2.