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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).✓ Correct
  2. B.If an abelian group G has property (P), then every element of G has finite order.✓ Correct
  3. C.The group Z\mathbb{Z} has property (P).
  4. D.If an abelian group G has property (P), then no element has order p2p^{2}, where p is a prime number.✓ Correct

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, and why this one tests property not inherited.

See pricing

50 are analysed free — try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

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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