3

My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP. The obvious representation using relations (see example for $B(3,3)$ below) is quite complex, because obtaining relations for large n is not an easy task.

f := FreeGroup(3);;
a := f.1;;
b := f.2;;
c := f.3;;
rels := [a^3, b^3, c^3, (a*b)^3, (a*c)^3, (b*c)^3, (a^2*b)^3, (a^2*c)^3, (b^2*c)^3, (a*b*c)^3, (a^2*b*c)^3, (a*b^2*c)^3, (b^2*a^2*c)^3];;
g := f / rels;;
  1. Is there a convenient way in GAP (maybe using some external package) to represent relatively free groups, in particular for Burnside groups, in particular for $B(n, 3)$?
  2. Is there a convenient way in GAP (maybe using some external package) to represent verbal subgroups (this can be used to simplify the method with relations mentioned above)?
arthurbesse
  • 149
  • Please ask one question at a time. – Shaun Apr 02 '24 at 20:29
  • @Shaun will keep in mind in for future, thanks. – arthurbesse Apr 02 '24 at 20:33
  • 3
    You could use the ANUPQ package applied to a free group of rank $n$ with the prime $3$ and the "exponent" option to compute $B(n,3)$. – Derek Holt Apr 02 '24 at 21:57
  • 3
    Just to emphasize Derk Holt's answer. This is exactly what the ANUPQ was built for. There isn't any setting in GPA that would have a similar effect apart from calling the ANUPQ from the system. – ahulpke Apr 03 '24 at 01:57

0 Answers0