CoCalc -- C6-Cases.ipynb

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Kernel: SageMath 9.6

Here we check that the ideal $I$ defined in the proof of Proposition 10.17 indeed fully realizes $G$ for $G = \mathbf{C}_2 \times \mathbf{C}_6$ (SmallGroup(12,5)), $G = \mathbf{C}_2 \times \mathbf{A}_4$ (SmallGroup(24, 13)), and $G = \mathbf{C}_2^4 \rtimes \mathbf{C}_6$ (SmallGroup(96,70)).

In [1]:

%run -i FPENAG.py

G = gap.SmallGroup(12,5)
[ag, bg, cg] = G.Generators()
xg = bg*cg
kG = gap.GroupRing(GF(2), G)
f = gap.Embedding(G, kG)
id = gap.Image(f, xg^0)

print("G = " + str(gap.StructureDescription(G)) + " of order " + str(G.Order()) + '.')
print()

x = gap.Image(f, xg)
a = gap.Image(f, ag)

# Here we define the subgroups V = A : <x^3> and W = A : <x^2> (see the proof).

V = G.Subgroup([ag, xg^3])
W = G.Subgroup([ag, xg^2])

print("V = " + str(gap.StructureDescription(V)) + " of order " + str(V.Order()) + '.')
print("W = " + str(gap.StructureDescription(W)) + " of order " + str(W.Order()) + '.')

# Here we construct the ideal given in the proof of the proposition
# that we claim fully realizes G.

Jgens = []
for vg in V.Elements():
    for wg in W.Elements():
        for ep in [0,1]:
            v = gap.Image(f, vg)
            w = gap.Image(f, wg)
            elt = id + v + w*x^(3*ep) + x^(3*ep)*v*x^(-3*ep)*w*x^(3*ep)
            if elt not in Jgens:
                Jgens += [elt]
J = gap.Ideal(kG, Jgens)

# Switch to a smaller set of generators, but check
# that we're dealing with exactly the same ideal as J above.

Igens = [id + a + x + a*x, id + x^3 + x + x^4]
I = gap.Ideal(kG, Jgens)

print()
print("I == J", I == J)
print()

kGmodI = kG/I

# Check that the ideal I defined in the proof of the proposition
# indeed fully realizes G.

analyzekGmodI(G, f, id, I, kGmodI, Igens, skipFRCheck=False)

Out[1]:

G = C6 x C2 of order 12.

V = C2 x C2 of order 4.
W = C6 of order 6.

I == J True

G = C6 x C2 of order 12.
G embeds in kG/I.
kG/I has 32 elements and 12 units.
G is fully realizable.

In [2]:

G = gap.SmallGroup(24,13)

print("G = " + str(gap.StructureDescription(G)) + " of order " + str(G.Order()) + '.')
print()

# Orders: 2, 3, 2, 2
[cg, dg, ag, bg] = G.Generators()

xg = cg*dg

V = G.Subgroup([ag, bg, xg^3])
W = G.Subgroup([ag, bg, xg^2])

print("V = " + str(gap.StructureDescription(V)) + " of order " + str(V.Order()) + '.')
print("W = " + str(gap.StructureDescription(W)) + " of order " + str(W.Order()) + '.')

kG = gap.GroupRing(GF(2), G)
f = gap.Embedding(G, kG)
id = gap.Image(f, xg^0)
x = gap.Image(f, xg)
a = gap.Image(f, ag)
b = gap.Image(f, bg)

Jgens = []
for vg in V.Elements():
    for wg in W.Elements():
        for ep in [0,1]:
            v = gap.Image(f, vg)
            w = gap.Image(f, wg)
            elt = id + v + w*x^(3*ep) + x^(3*ep)*v*x^(-3*ep)*w*x^(3*ep)
            if elt not in Jgens:
                Jgens += [elt]
J = gap.Ideal(kG, Jgens)

Igens = [id + b + x^2 + b*x^2, id + a + x^2 + a*x^2, id + x + x^3 + x^4]
I = gap.Ideal(kG, Igens)
print()
print("I == J", I == J)

kGmodI = kG/I

analyzekGmodI(G, f, id, I, kGmodI, Igens, skipFRCheck=False)

Out[2]:

G = C2 x A4 of order 24.

V = C2 x C2 x C2 of order 8.
W = A4 of order 12.

I == J True
G = C2 x A4 of order 24.
G embeds in kG/I.
kG/I has 64 elements and 24 units.
G is fully realizable.

In [3]:

%run -i FPENAG.py

G = gap.SmallGroup(96,70)

print("G = " + str(gap.StructureDescription(G)) + " of order " + str(G.Order()) + '.')
print()

# Orders: 2, 3, 2, 2, 2, 2.
[x3g, x2g, ag, gg, bg, dg] = G.Generators()

xg = x3g*x2g

kG = gap.GroupRing(GF(2), G)
f = gap.Embedding(G, kG)
id = gap.Image(f, ag^0)

[x, a, b, g, d] = [gap.Image(f, t) for t in [xg, ag, bg, gg, dg]]

# x has order 6, <a, b, g, d> is C_2^4.
# G = C_2^4 : C_6, C_6 = <x>.
#
# The action t |---> x*t*x^(-1) maps
#
#     a |---> a b g d
#     b |---> b d
#     g |---> a b
#     d |---> b

print("Checking action identities:")
print(x*a*x^(-1) == a*b*g*d, x*b*x^(-1) == b*d, x*g*x^(-1) == a*b, x*d*x^(-1) == b)

print()

V = G.Subgroup([ag, bg, gg, dg, xg^3])
W = G.Subgroup([ag, bg, gg, dg, xg^2])

print("V = " + str(gap.StructureDescription(V)) + " of order " + str(V.Order()) + '.')
print("W = " + str(gap.StructureDescription(W)) + " of order " + str(W.Order()) + '.')

Jgens = []
for vg in V.Elements():
    for wg in W.Elements():
        for ep in [0,1]:
            v = gap.Image(f, vg)
            w = gap.Image(f, wg)
            elt = id + v + w*x^(3*ep) + x^(3*ep)*v*x^(-3*ep)*w*x^(3*ep)
            Jgens += [elt]
J = gap.Ideal(kG, Jgens)

g1 = id + x^2+ x^3 + x^5
g2 = id + x^2 + a + a* x^2

Igens = [g1, g2]
I = gap.Ideal(kG, Igens)
kGmodI = kG/I

print()
print("I == J", I == J)
print()
analyzekGmodI(G, f, id, I, kGmodI, Igens, skipFRCheck=False)

Out[3]:

G = ((C2 x C2 x C2 x C2) : C2) : C3 of order 96.

Checking action identities:
True True True True

V = (C2 x C2 x C2 x C2) : C2 of order 32.
W = (C2 x C2 x C2 x C2) : C3 of order 48.

I == J True

G = ((C2 x C2 x C2 x C2) : C2) : C3 of order 96.
G embeds in kG/I.
kG/I has 256 elements and 96 units.
G is fully realizable.

In [0]:

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