Notes

//CALISAN DENEME DOSYASI
ZZ:=Integers();
p:=5; /* can be obtained from G */
n:=3; /* can be obtained from G */
i:=0;
basamak:= 0;
gl:=GL(n,p);
cptone := 0;
cptwo := 0;


devam:=true;

while devam do

i+:=1;
// both g1 and g2 should not be nonmodular! Wlog, take g1 modular
g1:=Random(gl); // g1 may be restricted to non/modular elements
g2:=Random(gl); // " " " "

if Order(g1) mod p eq 0 and IsUpperTriangular(g1) then

G:=MatrixGroup<n,GF(p)|g1,g2>;


if #G ne #gl then


R:=InvariantRing(G);

mylist:=[g : g in G | Order(g) mod p ne 0];
h_gen:=[];

while #mylist ne 0 do
mm,mn:=Max([Order(g) : g in mylist]);
g:=mylist[mn];
Append(~h_gen,g);
eliminate_g:=[g^i : i in [1..mm]];
for g in mylist do
if g in eliminate_g then Exclude(~mylist, g); end if;
end for;
end while;
cptone +:= 1;
"checkpoint 1", cptone;

H:=sub<G|h_gen>;
cptwo +:= 1;
"checkpoint 2", cptwo;

if #h_gen/#G le 1/p then
if #H lt #G then devam:=false; print i;


K:=InvariantRing(H);
P<[x]>:=PolynomialRing(R) ;

QG,phi:=quo<G|H>;

myS:=[g : g in G | g notin H and #Factorization(Order(g)) eq 1];
p_gen:=[];

while #myS ne 0 do
mm,mn:=Max([Order(g) : g in myS]);
g:=myS[mn];
gp:=g^(ZZ!(mm/p)); /* Now, gp is of Order p */
Append(~p_gen,gp);
im_g:=[phi(g^i) : i in [1..mm]];
for g in myS do
if phi(g) in im_g then Exclude(~myS, g); end if;
end for;
end while;




/* MAIN CODE */


kmax:=Max([TotalDegree(f) : f in PrimaryInvariants(R)]);
for k in [1..kmax] do /* DIKKAT XXXOOO */

K_k:=InvariantsOfDegree(K,k); /* Vector space of H-invariants of degree k */

if #K_k gt 0 then
/*
fullVS - vector space of all polynomial ring of degree k
*/

d_k:=n+#K_k;
mon_list:=IndexedSetToSequence(MonomialsOfDegree(P,k));
Sort(~mon_list); /* smallest first */
Reverse(~mon_list); /* largest first! IMPORTANT: This order is required for the below code */
all_dk:=#mon_list;

/* Vector space should be defined here */

fullVS:=VectorSpace(GF(p),all_dk);

hinv:=[]; /* initialize */

for f in K_k do
vf:=[];
cf,mf:=CoefficientsAndMonomials(f);

j:=1;
for u in mon_list do
if #mf ge j then
if u eq mf[j] then
Append(~vf,cf[j]); j+:=1;
else
Append(~vf,0);
end if;
else
Append(~vf,0);
end if;
end for; /* vf is the vector form of f */

Append(~hinv,vf);
end for;

genVS:=sub<fullVS|hinv>;

/*
Subspace of H-invariants is formed above.
Now, ginv will be constructed by generators from p_gen
*/

for g in p_gen do; /* DIKKAT XOX */

/*
Group action is extended to new coefficients
*/
g_k:=IdentityMatrix(GF(p),d_k);
for i,j in [1..n] do /* FXF */
g_k[i,j]:=g[i,j];
end for; /* FXF */

G_k:=MatrixGroup<d_k,GF(p) | g_k>;
R_k:=InvariantRing(G_k);
P_k:=PolynomialRing(R_k);

f:=0;
for i in [1..#K_k] do /* FO */
ff:=K_k[i];
ff:=eval Sprint(ff);
f+:=ff*P_k.(i+n);
end for; /* FO */

// f:=&+[(P_k!K_k[i])*P_k.(i+n) : i in [1..#K_k]]; /* f is in P_k */

df:=f;
for i in [1..p-1] do /* FFOO */
df:=df^g_k-df; /* df is also in P_k */
end for; /* FFOO */

Pb:=PolynomialRing(GF(p),#K_k);
Ps< [x]>:=PolynomialRing(Pb,3);

tt:=[x];
for i in [1..#K_k] do /* FFFFF */
tt:=Append(tt,Pb.i);
end for; /* FFFFF */

phi := hom< P_k -> Ps| tt>;

mycoef:=Coefficients(phi(df)); /* phi(df) is in Ps */
mycoef:=Append(mycoef,0); /* Check! Required? */

CoefMat:=[];
for f in mycoef do /* FF */
temp:=[];
for i in [1..#K_k] do /* FFF */
temp:=Append(temp,Coefficient(f,i,1));
end for; /* FFF */
CoefMat:=Append(CoefMat,temp);
end for; /* FF */
CoefMat:=Matrix(GF(p),CoefMat);
soln:=RowSequence(NullspaceMatrix(Transpose(CoefMat))); /* Check! Required? */

ginv_basis:={};
for j in [1..#soln] do /* FFFF */
f:=&+[K_k[i]*soln[j][i] : i in [1..#K_k]];
Include(~ginv_basis,f);
end for; /* FFFF */

print "k: ", k, " ginv: ", ginv_basis;

ginvp:=[]; /* initialize */

for f in ginv_basis do /* OO */
vf:=[];
cf,mf:=CoefficientsAndMonomials(f);

j:=1;
for u in mon_list do /* OOO */
if #mf ge j then
if u eq mf[j] then
Append(~vf,cf[j]); j+:=1;
else
Append(~vf,0);
end if;
else
Append(~vf,0);
end if;
end for; /* Line 190 daki for OOO vf is the vector form of f */

Append(~ginvp,vf);
end for; /* Line 185teki for OO */

genVS:=genVS meet sub<fullVS|ginvp>;
end for; /* for g in p_gen */ /* DIKKATTTTTTT XOX */

if Dimension(genVS) ne #InvariantsOfDegree(G,k) then
print "This group does not provide co-ex, see k: ", k;
end if;

end if; /* #K_k > 0 */
end for; /* for k in [1..kmax] DIKKATTTT XXXOOO */

basamak :=+1;
g1 :=Random(gl);
end if;
end if;
end if;
end if;



end while;