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Part CCSIR NET December 2024the-property-passes-to-subgroups-so-test-cyclic-ones

The property passes to subgroups so test cyclic ones

An abelian group G is said to have property (P) if for any subgroup N of G, there exists a subgroup H of G such that G = N + H and N ∩ H = {0}. Which of the following statements are true?

  1. A.If an abelian group G has property (P), every subgroup of G has property (P).✓
  2. B.If an abelian group G has property (P), then every element of G has finite order.✓
  3. C.The group has property (P).
  4. D.If an abelian group G has property (P), then no element has order , where p is a prime number.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: (ℤ/2^kℤ) is cyclic for every k

More on this topic

The chapter behind this: Structure of finite abelian groups — free to read

From Groups › Finite abelian groups

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