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Part BCSIR NET December 2024cyclic-only-when-n-is-coprime-to-phi-n

Cyclic only when n is coprime to phi n

For integers n > 1, let G(n) denote the number of groups of order n, up to isomorphism, i.e. G(n) is the number of isomorphism classes of groups of order n. Which of the following statements is true?

  1. A.If G(n) = 1, then n is prime.
  2. B.G(8) = 2
  3. C.If gcd⁡(n,φ(n))>1\gcd(n, \varphi(n)) > 1, then G(n) > 1. (Here φ\varphi denotes the Euler φ−\varphi-function.)✓ Correct
  4. D.lim⁡sup⁡(n→∞)G(n)=2\lim \sup(n\to\infty) G(n) = 2

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 standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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