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?

A.If G(n) = 1, then n is prime.
B.G(8) = 2
C.If $g cd (n, φ (n)) > 1$ , then G(n) > 1. (Here $φ$ denotes the Euler $φ -$ function.)✓
D. $lim sup (n \to \infty) G (n) = 2$

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.

See pricing

50 are analysed free — try those first.

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

Cyclic only when n is coprime to phi n

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