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Part CCSIR NET December 2024closure-needs-the-elements-to-commute

Closure needs the elements to commute

Let G be a finite group. For any prime number p, let E(p) = {g ∈ G | g^p = 1}. Which of the following statements are true?

  1. A.If p divides |G|, then E(p) is a subgroup of G.
  2. B.For all p, E(p) is a subgroup of G.
  3. C.If E(p) is a subgroup of G, then p divides |G|.
  4. D.If G is abelian, then E(p) is a subgroup of G.✓ 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 hypothesis dropped.

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50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

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

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