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For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$.

I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the property, that there is no prime power $p^k|n$, such that $p^k\equiv 1\ (\ mod\ q\ )$ for some prime $q|n$, then every group of order $n$ is abelian and $n$ is called an abelian number.

In OEIS, I found a slightly different definition of abelian numbers. Is the criterion I mentioned correct ?

The number of abelian groups of order $n$ can be easily calculated (assuming the prime factorization of $n$ is known). But what is the situation for general cubefree numbers $n$ ?

Is the cubefree case easy enough that the number of groups can be efficiently calculated, or are there cubefree numbers $n$ (of course with known factorization), for which the number of groups of order $n$ is unknown ?

Olexandr Konovalov
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Peter
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  • Yes the condition you wrote down for an abelian number is correct. It is both necessary and sufficient. But perhaps you should say that $k=1$ or $2$ (you don't want to allow $k=0$). – Derek Holt Dec 02 '15 at 13:35
  • Is the number of groups of order $p^2q^2\ ,\ p<q$ bounded from above by a constant or depending on p ? – Peter Dec 02 '15 at 16:45
  • It must depend on $p$. The number of groups of order $pq^2$ where $p|q-1$ increases with $p$. (It is just over $(p-1)/2$. – Derek Holt Dec 02 '15 at 18:54
  • @Derek Holt how can we conclude that the number of groups of order $p^2q^2$ is at least the number of groups of order $pq^2$ ? – Peter Dec 02 '15 at 18:59
  • Can we take the direct product of any group of order $pq^2$ and $\mathbb Z_p$ ? – Peter Dec 02 '15 at 19:00
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    If $m|n$ then the number of groups of order $n$ is at least as large as the number of order $m$. As you say, you can use just take direct products with $C_{n/m}$. The Krull-Schmidt Theorem ensures that $G \times C_k \cong H \times C_k \Rightarrow G \cong H$. – Derek Holt Dec 02 '15 at 23:05

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You seem to be using "unknown" to mean that there is no simple and efficient algorithm to determine the groups of a particular order, which is not very precise.

But algorithms to determine this number have been devised and implemented.in GAP. See the final section of this Diploma Thesis by Heiko Dietrich.

Derek Holt
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