I know that all groups of order $\leq$ 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?

  • $\begingroup$ There is a simple formula for the number of ABELIAN groups of order $n$, so the abelian numbers are useful. The smallest example with two distinct prime factors is $15$. $\endgroup$ – Peter Dec 2 '15 at 17:49

Yes, such numbers are called abelian numbers.

A number $n$ is abelian, if and only if it is cube-free and there is no prime power $p^k\mid n$ with $k\ge 1$, such that $p^k\equiv1\pmod q$ for some prime $q\mid n$.

In particular, if $p$ is a prime, then $p$ and $p^2$ are abelian. The number $pq$ with $2<p<q$ is abelian, if and only if $p$ does not divide $q-1$. The only even abelian numbers are $2$ and $4$.

  • $\begingroup$ Note, that there is a similar definition for cyclic numbers (numbers $n$, such that every group of order $n$ is cyclic). A number $n$ is cyclic if and only if $gcd(n,\phi(n))=1$, where $\phi(n)$ if the Euler-Phi-Function. $\endgroup$ – Peter Dec 2 '15 at 17:40
  • $\begingroup$ Is there another name for these Abelian numbers? I can't seem to find anything on them. $\endgroup$ – This Play Name Dec 2 '15 at 17:40
  • $\begingroup$ "The only even abelian number is $2$." Apart from $4$. $\endgroup$ – TonyK Dec 2 '15 at 17:41
  • $\begingroup$ @TonyK yes, I mixed it with the cyclic numbers ... Everything OK, now ? $\endgroup$ – Peter Dec 2 '15 at 17:42

The terminology you're looking for is:

Definition. A number $n \in \mathbb{N}$ is said to be

  • cyclic iff all groups of order $n$ are cyclic.

  • abelian iff all groups of order $n$ are abelian.

  • nilpotent iff all groups of order $n$ are nilpotent.

Using basic group theory, we see that these are related as follows:

Proposition. For natural numbers, we have: $$\mbox{ prime } \rightarrow \mbox{ cyclic } \rightarrow \mbox{ abelian } \rightarrow \mbox{ nilpotent }$$

A Peter explains, your question is essentially: does there exist an abelian number strictly greater than $5$ that isn't prime? Yes, $15$ does the trick. In fact, $15$ has the stronger property of being a cyclic number strictly greater than $5$ that isn't prime.

You can have a look at this mathoverflow post for more information.


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