Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I do know of $3$ classes of groups (up to isomorphism) of order $24$ that are commutative (direct products of $\mathbb{Z}$/(factors of $24$)$\mathbb{Z}$. Can you just take the semi direct product instead of the direct product and make these groups non commutative?? Does this normally guarantee non-commutative groups?

share|cite|improve this question
up vote 2 down vote accepted

Taking your questions in order:

Yes, there exist noncommutative groups of order $24$: $S_4$ for example.

No, of course the fact that you know some commutative groups of order $n$ does not prove that every group of order $n$ is commutative: why would it?

Yes, a non-trivial semidirect product is always noncommutative. However, non-trivial products do not always exist. For example, there are none for $\Bbb{Z}/3\Bbb{Z}$ and $\Bbb{Z}/8\Bbb{Z}$ with $\Bbb{Z}/8\Bbb{Z}$ normal, but there are with $\Bbb{Z}/3\Bbb{Z}$ normal.

share|cite|improve this answer
In this case, I forgot the clear counterexample of $S_4$, as all of you have mentioned, but there are certain orders, where groups can be either commutative or non commutative, but not both. – chubbycantorset Dec 11 '12 at 21:00
@chubbycantorset: What? A group is always commutative or noncommutative, but not both. If you mean every group of a given order $n$, then note that it's impossible for every group of order $n$ to be noncommutative (cyclic groups exist). It is possible for every group of order $n$ to be commutative (e.g. when $n$ is a prime), but this certainly isn't implied by the fact that there are some commutative groups of order $n$. – Chris Eagle Dec 11 '12 at 21:04

$S_4$--the group of permutations on $4$ elements--is a non-commutative group of order $24$.

Another example is the dihedral group of order $24$, which can be obtained as a semidirect product of the cyclic groups of order $2$ and order $12$. (Recall that cyclic groups are abelian.)

share|cite|improve this answer
Ah, right. Of course. But can you construct non commutative groups from taking the semi direct products, instead of taking the direct product in the groups that I mentioned? – chubbycantorset Dec 11 '12 at 20:52
Yes, indeed. See my edit. – Cameron Buie Dec 11 '12 at 20:56

Oh, lots of them. In fact, up to isomorphism, there are $\,12\,$ different non-abelian groups of order $\,24\,$ , and perhaps the easiest one is $\,S_4\,$ , or the dihedral $\,D_{12}\,$ ...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.