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Part BCSIR NET December 2024the-union-of-conjugates-never-fills-the-group

The union of conjugates never fills the group

Let G be a group of order n > 3 and H be a subgroup with 1 < |H| < n. Consider the set X = ⋃(g∈G) gHg⁻. Which of the following statements is true?

  1. A.If G is abelian, then |X| = n.
  2. B.If |X| divides n, then G is abelian.
  3. C.|X| < n✓
  4. D.|X| divides 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: Every group of order p² is cyclic

More on this topic

The chapter behind this: Group actions, the class equation and p-groups — free to read

From Groups › Group actions, class equation, p-groups

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