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Part BCSIR NET December 2025count-divisors-of-p-to-the-n-minus-1-to-build-any-target-count

Count divisors of p to the n minus 1 to build any target count

For a finite group G, let S(G) denote the number of subgroups of G. Which of the following statements is necessarily true?

  1. A.Let G and G′ be finite groups such that S(G) = S(G′). Then G is isomorphic to G′.
  2. B.If S(G) = 4, then |G| = pᵐ for some prime number p and positive integer m.
  3. C.If S(G) = 5, then G is a cyclic group.
  4. D.For every positive integer n, there exists a finite group G such that S(G) = n.✓

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.

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50 are analysed free — try those first.

Related counterexample: If d divides |G| then G has a subgroup of order d

More on this topic

The chapter behind this: Lagrange, orders and cyclic groups — free to read

From Groups › Subgroups, cosets, Lagrange, cyclic groups

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