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## Groups of order p^2q^2 are solable by Burnside's pq-theorem.
## Let G be a group of order p^2q^2, then G has a normal Sylow subgroup unless |G| = 36, in which case the classification of G can be dealt with by direct computation.
## Let Q \in Syl_q(G), and P \in Syl_p(G). Since P and Q are abelian, G is nilpotent if and only if G is abelian, in which case the classification is given by FTFGAG.
## For non-nilpotent groups G with a normal Sylow subgroup, the classifcation of G follows from the classification of semidirect products Q \ltimes P and P \ltimes Q,
## where Q \in \{C_{q^2}, C_q^2\}, P \in \{C_{p^2}, C_p^2\}.
## For further details, see [2, Section 3.2 & 3.6].
############################################################################ all groups P2Q2
SOTRec.allGroupsP2Q2 := function(p, q)
local a, b, c, d, e, f, qq, ii, qqq, iii,
s1, S1, s2, S2, r1, R1, r2, R2, all, list, G, k, t, matq, matq2, matp, matp2;;
####
Assert(1, p > q);
Assert(1, IsPrimeInt(p));
Assert(1, IsPrimeInt(q));
#### Cluster 0: abelian groups
all := [ [ [p, p, q, q], [1, [2, 1]], [3, [4, 1]] ],
[ [p, p, q, q], [3, [4, 1]] ],
[ [p, p, q, q], [1, [2, 1]] ],
[ [p, p, q, q], [2, [3, 1]] ] ];
## Computing roots
a := Z(p); #\sigma_p
b := Z(q); #\sigma_q
#\sigma_{p^2}
c := ZmodnZObj(Int(Z(p)), p^2);
if not c^(p - 1) = ZmodnZObj(1, p^2) then
d := c;
else
d := c + 1;
fi;
#\sigma_{q^2}
e := ZmodnZObj(Int(Z(q)), q^2);
if not e^(q - 1) = ZmodnZObj(1, q^2) then
f := e;
else
f := e + 1;
fi;
if (p - 1) mod q = 0 then
#\rho(p, q)
s1 := a^((p-1)/q);
S1 := Int(s1);
#\rho(p^2, q)
r1 := d^(p*(p-1)/q);
R1 := Int(r1);
if (p - 1) mod (q^2) = 0 then
#\rho(p, q^2)
s2 := a^((p-1)/(q^2));
S2 := Int(s2);
#\rho(p^2, q)
r2 := d^(p*(p-1)/(q^2));
R2 := Int(r2);
fi;
fi;
#### Cluster 1: ##C_{q^2} \ltimes C_{p^2}
if (p - 1) mod q = 0 then
ii := R1 mod p;
qq := (R1 - ii)/p;
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]] ]);
##C_{q^2} \ltimes C_{p^2}, \phi(Q) = C_q
if (p - 1) mod q^2 = 0 then
ii := R2 mod p;
qq := (R2 - ii)/p;
iii := R1 mod p;
qqq := (R1 - iii)/p;
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]], [3, 2, [3, iii, 4, qqq]], [4, 2, [4, iii]] ]);
fi;
fi;
#### Cluster 2: C_q^2 \ltimes C_{p^2}
if (p - 1) mod q = 0 then
ii := R1 mod p;
qq := (R1 - ii)/p;
Add(all, [ [q, q, p, p], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]] ]);
fi;
#### Cluster 3: C_{q^2} \ltimes C_p^2 or C_9 \ltimes C_2^2
if (p = 3 and q = 2) then
matq := SOTRec.QthRootGL2P(2, 3);
Add(all, [ [3, 3, 2, 2], [1, [2, 1]],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (p - 1) mod q = 0 then
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S1]] ]);
if q mod 2 = 1 then
t := (q - 1)/2;
else
t := 0;
fi;
for k in [0..t] do
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S1]], [4, 1, [4, Int(a^(Int(b^k)*(p-1)/q))]] ]);
od;
if (p - 1) mod q^2 = 0 and q > 2 then
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S2]], [3, 2, [3, S1]] ]);
for k in [0..(q^2 - q)/2] do
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S2]], [4, 1, [4, Int((a^(Int(f^k)*(p-1)/(q^2))))]], [3, 2, [3, S1]], [4, 2, [4, Int((a^(Int(f^k)*(p-1)/q)))]] ]); ##C_{q^2} \ltimes C_p^2, \phi(Q) = C_{q^2}
od;
for k in [1..(q - 1)] do
Add(all, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S2]], [3, 2, [3, S1]], [4, 1, [4, Int(a^(k*(p-1)/q))]] ]);
od;
fi;
if (p - 1) mod q^2 = 0 and q = 2 then
ii := Int(d^((p^2-p)/4)) mod p;
qq := (Int(d^((p^2-p)/4)) - ii)/p;
Add(all, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int((Z(p))^((p - 1)/4))]], [3, 2, [3, p - 1]] ]);
Add(all, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int(a^((p - 1)/4))]], [3, 2, [3, p-1]], [4, 1, [4, Int(a^((p - 1)/4))]], [4, 2, [4, p-1]] ]);
Add(all, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int(a^((p - 1)/4))]], [3, 2, [3, p-1]], [4, 1, [4, Int(a^(3*(p - 1)/4))]], [4, 2, [4, p-1]] ]);
Add(all, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, p-1]], [4, 1, [4, Int(a^((p - 1)/4))]], [4, 2, [4, p-1]] ]);
fi;
fi;
if (p + 1) mod q = 0 and q mod 2 = 1 then
matq := SOTRec.QthRootGL2P(p, q);
Add(all, [ [q, q, p, p], [1, [2, 1]],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (p + 1) mod (q^2) = 0 then
matq2 := SOTRec.QsquaredthRootGL2P(p, q);
matq := matq2^q;
Add(all, [ [q, q, p, p], [1, [2, 1]],
[3, 1, [3, Int(matq2[1][1]), 4, Int(matq2[2][1])]],
[4, 1, [3, Int(matq2[1][2]), 4, Int(matq2[2][2])]],
[3, 2, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 2, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]); ##C_{q^2} \ltimes C_p^2, \phi(Q) = C_{q^2}
fi;
#### Cluster 4: C_q^2 \ltimes C_p^2 or C_3 \ltimes C_2^2
if (p - 1) mod q = 0 then
Add(all, [ [q, q, p, p], [3, 1, [3, S1]] ]);
if q mod 2 = 1 then
t := (q - 1)/2;
else
t := 0;
fi;
for k in [0..t] do
Add(all, [ [q, q, p, p], [3, 1, [3, S1]], [4, 1, [4, Int((a^(Int(b^k)*(p-1)/q)))]] ]);
od;
Add(all, [ [q, q, p, p], [3, 1, [3, S1]], [4, 2, [4, S1]] ]);
if p = 3 and q = 2 then
matq := SOTRec.QthRootGL2P(2, 3);
Add(all, [ [3, 3, 2, 2],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
fi;
if (p + 1) mod q = 0 and q > 2 then
matq := SOTRec.QthRootGL2P(p, q);
Add(all, [ [q, q, p, p],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ] );
fi;
list := List(all, x -> SOTRec.groupFromData(x));
return list;
end;
##
############################################################################
SOTRec.NumberGroupsP2Q2 := function(p, q)
local w;
####
Assert(1, p > q);
Assert(1, IsPrimeInt(p));
Assert(1, IsPrimeInt(q));
if p = 3 and q = 2 then
w := 14;
elif q = 2 then
w := 11 + 5*SOTRec.w((p-1), 4) + SOTRec.w((p+1), 4);
else
w := 4 + (6+q)*SOTRec.w((p-1), q) + (4+q+q^2)*SOTRec.w((p-1), q^2)/2 + 2*SOTRec.w((p+1),q) + SOTRec.w((p+1), q^2);
fi;
return w;
end;
############################################################################
SOTRec.GroupP2Q2 := function(p, q, i)
local a, b, c, d, e, f, qq, ii, qqq, iii, l0, lst, G, k, t, matq, matq2, matp, matp2,
c1, c2, c3, c4, c5, s1, S1, s2, S2, r1, R1, r2, R2;
####
Assert(1, p > q);
Assert(1, IsPrimeInt(p));
Assert(1, IsPrimeInt(q));
#### case1: q nmid (p-1), q nmid (p^2 -1), q > 2
l0 := [ [ [p, p, q, q], [1, [2, 1]], [3, [4, 1]] ], [ [p, p, q, q], [3, [4, 1]] ], [ [p, p, q, q], [1, [2, 1]] ], [ [p, p, q, q], [2, [3, 1]] ] ];
if i < 5 then
return SOTRec.groupFromData(l0[i]);
fi;
a := Z(p);
b := Z(q);
c := ZmodnZObj(Int(Z(p)), p^2);
if not c^(p - 1) = ZmodnZObj(1, p^2) then
d := c;
else
d := c + 1;
fi;
e := ZmodnZObj(Int(Z(q)), q^2);
if not e^(q - 1) = ZmodnZObj(1, q^2) then
f := e;
else
f := e + 1;
fi;
if (p - 1) mod q = 0 then
#\rho(p, q)
s1 := a^((p-1)/q);
S1 := Int(s1);
#\rho(p^2, q)
r1 := d^(p*(p-1)/q);
R1 := Int(r1);
if (p - 1) mod (q^2) = 0 then
#\rho(p, q^2)
s2 := a^((p-1)/(q^2));
S2 := Int(s2);
#\rho(p^2, q)
r2 := d^(p*(p-1)/(q^2));
R2 := Int(r2);
fi;
fi;
#### Enumeration
c1 := SOTRec.w((p - 1), q) + SOTRec.w((p - 1), q^2);
c2 := SOTRec.w((p - 1), q);
c3 := 1/2*(q + 3 - SOTRec.w(2, q))*SOTRec.w((p - 1), q) + 1/2*(q^2 + q + 2)*SOTRec.w((p - 1), q^2) + (1 - SOTRec.w(2, q))*SOTRec.w((p + 1), q) + SOTRec.w((p + 1), q^2) + SOTRec.delta(p^2*q^2, 36);
c4 := 1/2*(q + 5 - SOTRec.w(2, q))*SOTRec.w((p - 1), q) + (1 - SOTRec.w(2, q))*SOTRec.w((p + 1), q) + SOTRec.delta(p^2*q^2, 36);
#### Cluster 1: ##C_{q^2} \ltimes C_{p^2}
if i > 4 and i < 5 + c1 then
lst := [];
if (p - 1) mod q = 0 then
ii := R1 mod p;
qq := (R1 - ii)/p;
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]] ]);
##C_{q^2} \ltimes C_{p^2}, \phi(Q) = C_q
if (p - 1) mod q^2 = 0 then
ii := R2 mod p;
qq := (R2 - ii)/p;
iii := R1 mod p;
qqq := (R1 - iii)/p;
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]], [3, 2, [3, iii, 4, qqq]], [4, 2, [4, iii]] ]);
fi;
fi;
return SOTRec.groupFromData(lst[i - 4]);
#### Cluster 2: C_q^2 \ltimes C_{p^2}
elif i > 4 + c1 and i < 5 + c1 + c2 then
lst := [];
if (p - 1) mod q = 0 then
ii := R1 mod p;
qq := (R1 - ii)/p;
Add(lst, [ [q, q, p, p], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [4, 1, [4, ii]] ]);
fi;
return SOTRec.groupFromData(lst[i - 4 -c1]);
#### Cluster 3: C_{q^2} \ltimes C_p^2
elif i > 4 + c1 + c2 and i < 5 + c1 + c2 + c3 then
lst := [];
if p = 3 and q = 2 then ## ! C_{p^2} \ltimes C_q^2
matq := SOTRec.QthRootGL2P(2, 3);
Add(lst, [ [3, 3, 2, 2], [1, [2, 1]],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (p - 1) mod q = 0 then
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S1]] ]);
if q mod 2 = 1 then
t := (q - 1)/2;
else
t := 0;
fi;
for k in [0..t] do
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S1]], [4, 1, [4, Int(a^(Int(b^k)*(p-1)/q))]] ]);
od;
if (p - 1) mod q^2 = 0 and q > 2 then
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, Int(a^((p - 1)/(q^2)))]], [3, 2, [3, Int(a^((p - 1)/q))]] ]);
for k in [0..(q^2 - q)/2] do
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S2]], [4, 1, [4, Int((a^(Int(f^k)*(p-1)/(q^2))))]], [3, 2, [3, S1]], [4, 2, [4, Int((a^(Int(f^k)*(p-1)/q)))]] ]); ##C_{q^2} \ltimes C_p^2, \phi(Q) = C_{q^2}
od;
for k in [1..(q - 1)] do
Add(lst, [ [q, q, p, p], [1, [2, 1]], [3, 1, [3, S2]], [3, 2, [3, S1]], [4, 1, [4, Int(a^(k*(p-1)/q))]] ]);
od;
fi;
if (p - 1) mod q^2 = 0 and q = 2 then
ii := Int(d^((p^2-p)/4)) mod p;
qq := (Int(d^((p^2-p)/4)) - ii)/p;
Add(lst, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int((Z(p))^((p - 1)/4))]], [3, 2, [3, p - 1]] ]);
Add(lst, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int(a^((p - 1)/4))]], [3, 2, [3, p-1]], [4, 1, [4, Int(a^((p - 1)/4))]], [4, 2, [4, p-1]] ]);
Add(lst, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, Int(a^((p - 1)/4))]], [3, 2, [3, p-1]], [4, 1, [4, Int(a^(3*(p - 1)/4))]], [4, 2, [4, p-1]] ]);
Add(lst, [ [2, 2, p, p], [1, [2, 1]], [3, 1, [3, p-1]], [4, 1, [4, Int(a^((p - 1)/4))]], [4, 2, [4, p-1]] ]);
fi;
fi;
if (p + 1) mod q = 0 and q mod 2 = 1 then
matq := SOTRec.QthRootGL2P(p, q);
Add(lst, [ [q, q, p, p], [1, [2, 1]],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (p + 1) mod (q^2) = 0 then
matq2 := SOTRec.QsquaredthRootGL2P(p, q);
matq := matq2^q;
Add(lst, [ [q, q, p, p], [1, [2, 1]],
[3, 1, [3, Int(matq2[1][1]), 4, Int(matq2[2][1])]],
[4, 1, [3, Int(matq2[1][2]), 4, Int(matq2[2][2])]],
[3, 2, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 2, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]); ##C_{q^2} \ltimes C_p^2, \phi(Q) = C_{q^2}
fi;
return SOTRec.groupFromData(lst[i - 4 - c1 - c2]);
#### Cluster 4: C_q^2 \ltimes C_p^2
elif i > 4 + c1 + c2 + c3 and i < 5 + c1 + c2 + c3 + c4 then
lst := [];
if (p - 1) mod q = 0 then
Add(lst, [ [q, q, p, p], [3, 1, [3, S1]] ]);
if q mod 2 = 1 then
t := (q - 1)/2;
else
t := 0;
fi;
for k in [0..t] do
Add(lst, [ [q, q, p, p], [3, 1, [3, S1]], [4, 1, [4, Int((a^(Int(b^k)*(p-1)/q)))]] ]);
od;
Add(lst, [ [q, q, p, p], [3, 1, [3, S1]], [4, 2, [4, S1]] ]);
if p = 3 and q = 2 then
matq := SOTRec.QthRootGL2P(2, 3);
Add(lst, [ [3, 3, 2, 2],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
fi;
if (p + 1) mod q = 0 and q > 2 then
matq := SOTRec.QthRootGL2P(p, q);
Add(lst, [ [q, q, p, p],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ] );
fi;
return SOTRec.groupFromData(lst[i - 4 - c1 - c2 - c3]);
fi;
end;
############################################################################
[ Dauer der Verarbeitung: 0.36 Sekunden
(vorverarbeitet)
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