CoCalc -- Q8.ipynb

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$\mathbf{Q}_8 = \langle x, y \, | \, x^4 = 1, x^2 = y^2, yx = x^{-1}y\rangle$ is not fully realizable.

In [1]:

%run -i FPENAG.py

G = gap.QuaternionGroup(8)

# Using the above notation, x is xg and y is yg.

[xg,yg,junk] = G.Generators()

gens = [xg, yg]
kG = gap.GroupRing(GF(2), G)
f = gap.Embedding(G,kG)

# x, y below are the generators as elements of
# the group algebra. "id" is the "1" in k[G].

[x, y] = [gap.Image(f, t) for t in gens]
id = gap.Image(f, xg^0)

In [2]:

# Check that our generators align with the notation above.

xg^4 == xg^0, xg^2 == yg^2, yg*xg == xg^(-1)*yg

Out[2]:

(True, True, True)

In [3]:

# This is just an example of a ring (kGmodI) that realizes,
# but does not fully realize, Q_8.

g1 = id + x + y + x*y
Igens = [g1]
I = gap.Ideal(kG, Igens)
kGmodI = kG/I

analyzekGmodI(G, f, id, I, kGmodI, Igens)

Out[3]:

G = Q8 of order 8.
G embeds in kG/I.
kG/I has 16 elements and 8 units.
kG/I realizes but does not fully realize G.

In [4]:

# Here, GAP gives an endomorphism of Q_8 that does not
# preserve the ideal generator 1 + x + y + xy. It's the
# endomorphism that sends x to x and y to xy; this
# endomoprhism sends 1 + x + y + xy to 1 + x + xy + x^2 y,
# which is not in the ideal.

preservesList(f, G, I, Igens, 'end', True)

Out[4]:

The  endomorphism
Pcgs([ x, y, y2 ]) -> [ x, x*y, y2 ]
sends element 1:
(Z(2)^0)*<identity> of ...+(Z(2)^0)*x+(Z(2)^0)*y+(Z(2)^0)*x*y
to the element:
(Z(2)^0)*<identity> of ...+(Z(2)^0)*x+(Z(2)^0)*x*y+(Z(2)^0)*y*y2
which is outside the ideal.

False

In [5]:

id + x + y + x^2*y in I

Out[5]:

False

Now we show that $G = \mathbf{Q}_8$ is not fully realizable.

In the group algebra $\mathbf{F}_2[G]$ , $\alpha$ is a unit if and only if it has augmentation 1 since $G$ is a finite 2-group. Hence, $1 + x + y$ is a unit. If $\mathbf{F}_2[G]/I$ realizes $G$ , then we must have $1 + x + y + u \in I$ for some $u \in G$ . We check each possible case below; in all cases the conclusion is that the ring does not fully realize $G$ . Either the ring realizes $G$ but does not fully realize it, or $G$ does not embed in $\mathbf{F}_2[G]/I$ (and hence would not embed in $\mathbf{F}_2[G]/J$ for any $J \supseteq I$ ).

In [6]:

for g in G.Elements():
    u = gap.Image(f, g)
    gen = id + x + y + u
    J = gap.Ideal(kG, [gen])
    Q = kG/J
    analyzekGmodI(G, f, id, J, Q, [gen], skipFRCheck=False)
    print()

Out[6]:

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G embeds in kG/I.
kG/I has 16 elements and 8 units.
kG/I realizes but does not fully realize G.

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G does not embed in kG/I.

G = Q8 of order 8.
G embeds in kG/I.
kG/I has 16 elements and 8 units.
kG/I realizes but does not fully realize G.

In [0]:

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