[GAP Forum] Free Vertices

[Siddiqua Mazhar]

Dear Sir/Madam,

Let x and y be a finite permutation representation,

x:=((6,10,12,14,9)(4,7,11,13,8));

y:=((1,3,6,9,5,2,4)(8,13,14,10,11,7,12));

I want to find the sequence C here from these permutations as,

C=[(1,3),(5,2)];

Here C are free vertices that means the points that are fixed under x and moved for y and are consecutive to each other. Let say 1 is fixed for x and 1^y=3 and then 3 is again fixed for x so we will add it in sequence c that i used in program c:=[j] and again 3^y=6 but 6 is not fixed for x so the loop will terminate. further I want to check other moved points of y that are free for x. Here 5 is fixed for x and moved for y, i.e. 5^y=2 and 2^x=2 so C will be C:=[5,2] because 2^y=4 and (4^x)<>4.

so here 5 and 2 are consective points and 1 and 3 are consecutive points.

this is what I have written in my previous email that 1 and 3 are fixed under x permutation and they are in y cylce next to each other, similarly, 5 and 2 are fixed under x and in y cycle they are next to each other moreover all of its subsequences.

For this I tried this program in GAP that gives me only C:=[5,2] ,indeed i want to calculate other value [1,3] by doing the same program.

The program I made as follows

FindSequenceOfFreeVertices:=function(x,y)
local j,c,next;

for j in MovedPoints(y) do
  if j^x=j then
  c:=[j];
  next:=j^y;



while next^x=next do
  Add(c,next);
  next:=next^y;
  od;
fi;
od;


Print("sequence of Free Vertices =",c,"\n");
end;

I hope this program will be easier for you to understand my question. Thanks