Same presentation groups,

Question

I am trying to look in GAP if it tells us that two groups have same presentation, like, IsIdenticalPresentation(G,H); but I couldn't find any. Could you please help me to find out, if there is any? In magma there is such command, but I am wondering if I could find in GAP as well.

Answer 1

There is no built-in function that does so, but one can do easily by mapping relators. The following function does so with minimum amount of cleverness as far as generator names, and arrangement of relators is concerned. Clearly even minimal changes will result in the presentations to be considered different.

IsIdenticalPresentation:=function(G,H)
local FG,FH,sh,perm,r;
  FG:=FreeGeneratorsOfFpGroup(G);
  FH:=FreeGeneratorsOfFpGroup(H);
  if Length(FG)<>Length(FH) then
    return false;
  fi;
  sh:=List(FH,String);
  perm:=List(FG,x->Position(sh,String(x)));
  if ForAny(perm,x->x=fail) then
    Info(InfoWarning,1,"Generator names differ, map 1<->1");
    perm:=[1..Length(FH)];
  fi;
  r:=List(RelatorsOfFpGroup(G),x->MappedWord(x,FG,FH{perm}));
  if Length(r)=Length(RelatorsOfFpGroup(H))
    and ForAll(r,x->x in RelatorsOfFpGroup(H)) then
    return true;
  else
    return false;
  fi;
end;