Skip to content
Part CCSIR NET December 2025there-are-only-five-groups-of-order-8-check-which-ones-actually-have-no-order-4-elements

There are only five groups of order 8 check which ones actually have no order 4 elements

Let G be a group of order 8. Which of the following statements are necessarily true?

  1. A.If there are no elements of order 4 in G, then G is abelian.✓
  2. B.If there are exactly two elements of order 4 in G, then G is abelian.
  3. C.If there are exactly six elements of order 4 in G, then G is abelian.
  4. D.There is an element of order 4 in G.

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: A group of order pq (p < q) is always cyclic

More on this topic

The chapter behind this: Sylow theorems and classifying small groups — free to read

From Groups › Sylow theorems and groups of small order

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

ShareWhatsAppTelegram