3
$\begingroup$

OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to calculate the number of groups (unless it is too large), but unfortunately, I do not have access to maple.

Does anyone know an online-calculator for the number-of-group-function, or a table with the known numbers upto $10,000$ or more ?

Or alternatively, allows PARI/GP, at least in principle, to calculate the number of groups of order $n$ ? I programmed the case $n$ squarefree, but I have no clue how to manage arbitary numbers $n$.

$\endgroup$
  • $\begingroup$ There isn't any general formula or algorithm known for computing the number of groups of a given order $n$, other than to construct all of them. Maple uses formulas for certain special cases, depending on the factorisation of $n$ (for instance, for square-free $n$, or for small powers of primes, etc.) and uses a table of known values for small $n<50000$. I expect GAP and Magma have something similar, but I don't know that PARI has that kind of functionality. $\endgroup$ – James Dec 1 '15 at 14:27
  • $\begingroup$ @James I was just curious. By the way, in the answer below, it is claimed that the number can be calculated in the case $p^2 q^2$. Do you know the formula ? $\endgroup$ – Peter Dec 1 '15 at 17:01
  • $\begingroup$ It's my understanding that some sort of formula or description is known, but I do not know it! I've never been able to get my hands on the papers where it is described. $\endgroup$ – James Dec 1 '15 at 20:33
  • $\begingroup$ For four primes, the situation is as follows. The case $p^4$ and the squarefree case are well-known. The case $p^2qr$ was done by Glenn (1906) ams.org/journals/tran/1906-007-01/S0002-9947-1906-1500737-3/… As far as I can see, the number of isomorphism type is not explicitly stated, but perhaps could be extracted with a little more work. $\endgroup$ – verret Dec 2 '15 at 6:24
  • $\begingroup$ The case $p^3q$ was done by Western (1898) plms.oxfordjournals.org/content/s1-30/1/209 His count is at the end. 15 groups when $q=2$. 6 or 19 groups when $q=3$ and $p$ odd. (For $p=2$, there are $15$.) etc... Take all of this with a grain of salt, for example, according to magma, there are $7$ groups of order $3\cdot 5^3$. Moreover, his count in the table is also off for 189 and 351. $\endgroup$ – verret Dec 2 '15 at 6:24
4
$\begingroup$

For the list of $n$'s which are included in the SmallGroups library, see : https://magma.maths.usyd.edu.au/magma/handbook/text/727

There are many missing n's which could be computed, even by hand (when $n=p^2q^2$, for example), but there are definitely some numbers less than $10000$ that are out of reach.

For example, the number of groups of order $2048$ is not actually known, higher powers of $2$ even less so, see for example https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf

(This is a good reference for your question in general.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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