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Finish unitary geometric maximals
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Co-authored-by: maxhauck <[email protected]>
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@ssiccha
ssiccha and maxhauck committed Oct 5, 2021
1 parent d437a95 commit 0f714ca
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175 changes: 168 additions & 7 deletions gap/ClassicalMaximals.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -198,13 +198,17 @@ C7SubgroupsSpecialLinearGroupGeneric := function(n, q)
highestPowern := Gcd(factorisationOfnExponents);

divisorsHighestPowern := DivisorsInt(highestPowern);

for t in divisorsHighestPowern{[2..Length(divisorsHighestPowern)]} do
m := RootInt(n, t);
if m < 3 then
continue;
fi;
tensorInducedSubgroup := TensorInducedDecompositionStabilizerInSL(m, t, q);
# Cf. Tables 3.5.A and 3.5.G in [3]
numberOfConjugates := Gcd(q - 1, m ^ (t - 1));
if m mod 4 = 2 and t = 2 and q mod 4 = 3 then
numberOfConjugates := QuoInt(numberOfConjugates, 2);
numberOfConjugates := Gcd(q - 1, m) / 2;
fi;
result := Concatenation(result,
ConjugatesInGeneralGroup(tensorInducedSubgroup,
Expand Down Expand Up @@ -455,7 +459,7 @@ C2SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
result := List(divisorListOfn, t -> SUNonDegenerateImprimitives(n, q, t));
# type GL(n / 2, q ^ 2).2 subgroups
if IsEvenInt(n) then
Add(result, SUIsotropicImprimitives);
Add(result, SUIsotropicImprimitives(n, q));
fi;

return result;
Expand Down Expand Up @@ -488,6 +492,74 @@ C4SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
return result;
end;

C3SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
return List(Filtered(PrimeDivisors(n), IsOddInt),
s -> GammaLMeetSU(n, q, s));
end;

C5SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
local factorisation, p, e, generatorGUMinusSU, primeDivisorsOfe,
degreeOfExtension, f, subfieldGroup, numberOfConjugates, result, epsilon;

factorisation := PrimePowersInt(q);
p := factorisation[1];
e := factorisation[2];
generatorGUMinusSU := GU(n, q).1;
primeDivisorsOfe := PrimeDivisors(e);

result := [];
# type GU subgroups
for degreeOfExtension in primeDivisorsOfe do
if IsEvenInt(degreeOfExtension) then
continue;
fi;
f := QuoInt(e, degreeOfExtension);
subfieldGroup := SubfieldSL(n, p, e, f);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := Gcd(n, QuoInt(q + 1, p ^ f + 1));
result := Concatenation(result,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
od;

# type GO subgroups
if IsOddInt(q) then
if IsOddInt(n) then
subfieldGroup := OrthogonalSubfieldSU(0, n, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := Gcd(n, q + 1);
result := Concatenation(result,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
else
for epsilon in [-1, 1] do
subfieldGroup := OrthogonalSubfieldSU(epsilon, n, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := QuoInt(Gcd(q + 1, n), 2);
result := Concatenation(result,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
od;
fi;
fi;

# type Sp subgroups
if IsEvenInt(n) then
subfieldGroup := SymplecticSubfieldSU(n, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := Gcd(QuoInt(n, 2), q + 1);
result := Concatenation(result,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
fi;

return result;
end;

C6SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
local factorisationOfq, p, e, factorisationOfn, r, m, result,
generatorGUMinusSU, numberOfConjugates, extraspecialNormalizerSubgroup;
Expand Down Expand Up @@ -521,7 +593,7 @@ C6SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
fi;
elif m >= 2 then
# n = 2 ^ m >= 4
if e = 1 and (q - 1) mod 4 <> 0 then
if e = 1 and 2 * e = OrderMod(p, 4) then
extraspecialNormalizerSubgroup := ExtraspecialNormalizerInSU(2, m, q);
# Cf. Tables 3.5.A and 3.5.G in [3]
numberOfConjugates := Gcd(n, q + 1);
Expand All @@ -538,10 +610,46 @@ C6SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
return result;
end;

C7SubgroupsSpecialUnitaryGroupGeneric := function(n, q)
local m, t, factorisationOfn, factorisationOfnExponents, highestPowern,
result, divisorsHighestPowern, numberOfConjugates, tensorInducedSubgroup,
generatorGUMinusSU;

result := [];
generatorGUMinusSU := GU(n, q).1;
factorisationOfn := PrimePowersInt(n);
# get all exponents of prime factorisation of n
factorisationOfnExponents := factorisationOfn{Filtered([1..Length(factorisationOfn)],
IsEvenInt)};
# n can be written as k ^ highestPowern with k an integer and highestPowern
# is maximal with this property
highestPowern := Gcd(factorisationOfnExponents);

divisorsHighestPowern := DivisorsInt(highestPowern);
for t in divisorsHighestPowern{[2..Length(divisorsHighestPowern)]} do
m := RootInt(n, t);
if (m = 3 and q = 2) or m < 3 then
continue;
fi;
tensorInducedSubgroup := TensorInducedDecompositionStabilizerInSU(m, t, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := Gcd(q + 1, m ^ (t - 1));
if m mod 4 = 2 and t = 2 and q mod 4 = 1 then
numberOfConjugates := Gcd(q + 1, m) / 2;
fi;
result := Concatenation(result,
ConjugatesInGeneralGroup(tensorInducedSubgroup,
generatorGUMinusSU,
numberOfConjugates));
od;

return result;
end;

InstallGlobalFunction(MaximalSubgroupClassRepsSpecialUnitaryGroup,
function(n, q, classes...)
local maximalSubgroups;
local maximalSubgroups, subfieldGroup, numberOfConjugates,
generatorGUMinusSU;

if Length(classes) = 0 then
classes := [1..9];
Expand All @@ -562,6 +670,7 @@ function(n, q, classes...)
Error("PSU(3, 2) is soluble");
fi;

generatorGUMinusSU := GU(n, q).1;

maximalSubgroups := [];

Expand Down Expand Up @@ -595,15 +704,29 @@ function(n, q, classes...)
# q = 2
Add(maximalSubgroups, SUNonDegenerateImprimitives(n, q, 4));
fi;
else
# n = 6 and q = 2
elif n = 6 and q = 2 then
# Cf. Theorem 6.3.10 in [1]
Add(maximalSubgroups, SUNonDegenerateImprimitives(n, q, 3));
Add(maximalSubgroups, SUNonDegenerateImprimitives(n, q, 2));
Add(maximalSubgroups, SUIsotropicImprimitives(n, q));
fi;
fi;

if 3 in classes then
# Class C3 subgroups ######################################################
# Cf. Propositions 3.2.3 (n = 3), 3.3.4 (n = 4), 3.4.3 (n = 5),
# 3.5.5 (n = 6), 3.6.3 (n = 7), 3.7.5 (n = 8),
# 3.8.3 (n = 9), 3.9.5 (n = 10), 3.10.3 (n = 11),
# 3.11.7 (n = 12) in [1]
if not (n = 6 and q = 2) and not (n = 3 and q = 5)
and not (n = 3 and q = 3)
and not (n = 5 and q = 2) then
maximalSubgroups := Concatenation(maximalSubgroups,
C3SubgroupsSpecialUnitaryGroupGeneric(n, q));
fi;
# There are no maximal C3 subgroups in the cases excluded above, cf.
# Proposition 3.5.5 and Theorem 6.3.10 in [1]
fi;

if 4 in classes then
# Class C4 subgroups ######################################################
# Cf. Propositions 3.5.6 (n = 6), 3.7.7 (n = 8), 3.9.6 (n = 10),
Expand All @@ -612,6 +735,37 @@ function(n, q, classes...)
C4SubgroupsSpecialUnitaryGroupGeneric(n, q));
fi;

if 5 in classes then
# Class C5 subgroups ######################################################
# Cf. Propositions 3.2.4 (n = 3), 3.3.5 (n = 4), 3.4.3 (n = 5),
# 3.5.7 (n = 6), 3.6.3 (n = 7), 3.7.8 (n = 8),
# 3.8.4 (n = 9), 3.9.7 (n = 10), 3.10.3 (n = 11),
# 3.11.9 (n = 12) in [1]
if not (n = 3 and q = 3) and not (n = 3 and q = 5) and not (n = 4 and q = 3) then
maximalSubgroups := Concatenation(maximalSubgroups,
C5SubgroupsSpecialUnitaryGroupGeneric(n, q));
# There are no maximal C5 subgroups for n = 3 and q = 3 or n = 3 and q = 5,
# cf. Proposition 3.2.4 and Theorem 6.3.10 in [1]
elif n = 4 and q = 3 then
# type Sp
subfieldGroup := SymplecticSubfieldSU(n, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := 2;
maximalSubgroups := Concatenation(maximalSubgroups,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
# type GO-
subfieldGroup := OrthogonalSubfieldSU(-1, n, q);
# Cf. Tables 3.5.B and 3.5.G in [3]
numberOfConjugates := 2;
maximalSubgroups := Concatenation(maximalSubgroups,
ConjugatesInGeneralGroup(subfieldGroup,
generatorGUMinusSU,
numberOfConjugates));
fi;
fi;

if 6 in classes then
# Class C6 subgroups ######################################################
# Cf. Lemma 3.1.6 (n = 2) and Propositions 3.2.5 (n = 3), 3.3.6 (n = 4),
Expand All @@ -627,6 +781,13 @@ function(n, q, classes...)
fi;
fi;

if 7 in classes then
# Class C7 subgroups ######################################################
# Cf. Proposition 3.8.6 (n = 9) in [1]
# For all other n, class C7 is empty.
maximalSubgroups := Concatenation(maximalSubgroups,
C7SubgroupsSpecialUnitaryGroupGeneric(n, q));
fi;

return maximalSubgroups;
end);
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