Help:=proc(): print(`fix(pi), AvF(G) , kefel(pi,sig) , GenG(S) `): 

print(`Tr(n,i) `):
end:


#fix(pi): inputs a permutation pi and outputs the
#number of fixed points
fix:=proc(pi) local n,i,co:
n:=nops(pi):
 
 co:=0:

  for i from 1 to n do
    if pi[i]=i then
      co:=co+1:
    fi:
 od:

co:

end:


with(combinat):

#the average number of fixed points of G
AvF:=proc(G) local  co,pi:

add(fix(pi),pi in G)/nops(G):

end:



#kefel(pi,sig): pi times sig (as perms)
kefel:=proc(pi,sig) local i,n:

n:=nops(pi):

if nops(sig)<>n then
 RETURN(FAIL):
fi:

[ seq( sig[pi[i]],i=1..n)]:

end:





#GenG(S):Given a set if permutations S all of the
#same size, outputs the group of permutations
#generated by them
GenG:=proc(S) local G,NewGuys,pi,sig:


if nops({seq(nops(pi),pi in S)})<>1 then
 RETURN(FAIL):
fi:


G:=S:

NewGuys:=  {seq(seq(kefel(pi,sig) , pi in G), sig in S)} minus G:

while NewGuys<>{} do
G:=G union NewGuys:
NewGuys:=  {seq(seq(kefel(pi,sig) , pi in G), sig in S)} minus G:
od:
G:

end:


#Tr(n,i): the permutation that trasposes i and i+1 (and nothing else
#gets changed)
Tr:=proc(n,i) local j:

if i>=n then
 RETURN(FAIL):
fi:

[seq(j,j=1..i-1),i+1,i,seq(j,j=i+2..n)]:
end:


#SymC3(): the symmetry group of a 3D cube
#with the convention that the top is 1,
#the bottom is 2, the left wall is 3, the
#the right wall is 5, the front wall is 4
#and the back wall is 6
SymC3:=proc() local pi,sig:

pi:=[1,2,6,3,4,5]:

end: