2

I need to compute first cohomology group with coefficients in $(\mathbb{Z}/n\mathbb{Z})^m$ of specific finite groups. I reduced the computation of cocycles to the following problem: compute the kernel of a matrix $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$.

  1. How do I compute it in general? Is there some good algorithm?
  2. Is there a GAP function / package for that?

Thanks for your answers!

Paweł Piwek
  • 455
  • 1
    GAP seems to have a way to do this when $n$ is a prime power: https://www.gap-system.org/Manuals/doc/ref/chap24.html#X8593A5337D3B2C70. I'm not a GAP expert, but when I tried computing the null space of a matrix over $\mathbb{Z}/12\mathbb{Z}$, I got an error, so it may not be implemented for general $n$. – John Palmieri Aug 13 '19 at 00:31
  • Thanks @JohnPalmieri, yeah, I need to do it for all sorts of numbers, not just prime powers, but maybe I could compute it for all maximal prime powers dividing $n$ and then piece it together with Chinese Remainder Theorem. – Paweł Piwek Aug 13 '19 at 11:48
  • Indeed there isn't anything pre-cooked if $n$ is not a prime power, probably because the structure is a bit messy. I think one could construct it using the Hermite normal form (I.e. $\Z$-kernel) , but it requires more than just to reduce the $\Z$ kernel and I have not thought it through what the structure of the kernel as a set actually is. – ahulpke Aug 14 '19 at 03:28

1 Answers1

2

You can use the function BasisNullspaceModN for that (it is undocumented, unfortunately, though we could change that in the next version of GAP). See also my answer here, which also describes a function NullspaceModN implemented using BasisNullspaceModN

Max Horn
  • 1,832
  • Thanks for your answer @Max. Does the fact that the function is undocumented mean that I need to read it every time I start GAP? Atm my GAP (4.9.2) doesn't recognise it as a function. – Paweł Piwek Aug 21 '19 at 23:34
  • 1
    I was unclear: NullspaceModN is only defined in my other linked answer. I clarified my answer here now. – Max Horn Aug 21 '19 at 23:39