Mathematics 666
Homework (due Sep. 18)
A. Hulpke
√
0 1
i 0
),(
)⟩ be the quaternion group of order 8. (You can
−1 and G = ⟨(
−1 0
0 −i
create it in GAP as
8) Let i =
Group([ [ [ 0, 1 ], [ -1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ] ])
for example and ask for its Elements.)
a) Construct an irreducible representation of G over the real numbers, acting on a 4-dimensional
vectorspace V ≅ R4 .
(Hint: Use an R-basis of C to get an R-basis of C2 . To show that no 2-dimensional submodule exists,
consider images of a nonzero vector (a, b, c, d) in this subspace under different elements of G, and
show that they will yield a basis of at least a 3-dimensional subspace.)
b) Determine the endomorphism ring EndRG (V ).
(Hint: The elements of EndRG (V ) are 4 × 4 matrices that commute with the generators of G. Use
this to deduce conditions on their entires. Then show that every matrix fulfilling these conditions
commutes with G.)
c) By Schur’s lemma EndRG (V ) must be a division ring. Can you identify it?
11) (Examples to show the necessity of the conditions for Maschke’s theorem)
1 1
)⟩ ≤ GL2 (Q). Show that the QGsubmodule ⟨(0, 1)⟩ ≤ Q2 does not have a QGa) Let G = ⟨(
0 1
complement.
⎛ 1 1 0 ⎞ ⎛ 1 0 0 ⎞
b) Let G = ⟨⎜ 1 0 2 ⎟ , ⎜ 1 1 2 ⎟⟩ ≤ GL3 (3) and V = F33 .
⎝ 1 2 0 ⎠ ⎝ 2 2 2 ⎠
Show that W = ⟨(1, 0, −1), (0, 1, 0)⟩ ≤ V is an F3 G-submodule that has no F3 G-complement.
12)
⎛
⎜
Let F = F2 , m = ⎜
⎜
⎝
0
0
1
0
1
0
1
0
0
1
0
0
1
1
0
1
⎞
⎟
⎟ ∈ F4×4
and G = ⟨m⟩. Then ∣m∣ = 7 (you do not need to show
2
⎟
⎠
this).
Let V = F42 be the FG module obtained by the matrix action (called the natural module) and W =
⟨(0, 0, 0, 1)⟩ ≤ V be a 1-dimensional FG-submodule. (You do not need to show any of the statements
so far.)
Using the method from the proof of Maschke’s theorem, construct a FG-submodule S such that
V = W ⊕ S.
13) Let G be a finite group and F = C. What is the relation between the one-dimensional Crepresentations of G, and the one-dimensional C-representations of G/G ′ (G ′ , the derived subgroup
of G, is the smallest normal subgroup N ⊲ G such that G/N is abelian)?
14) Consider the group algebra A = F2 S3 for S3 over the
field with 2 elements.
a) Determine the submodules of A A. (Hint: the GAP
commands
RegularModule(G,GF(2))[2];
and
MTX.BasesSubmodules.)
b) Determine the isomorphism types of simple modules
(MTX.CollectedFactors).
c) Draw the submodule lattice for A A and identify the
different H S (A) and J(A). Hint: If b is a list of submodule
bases the following GAP commands below determine for
each submodule those that it is included in, and those in
which it is a maximal submodule.
Compare with the structure for the submodule lattice for
F3 S3 displayed at the right.
GF(3)S3
6
28
5
26
4
3
2
1
0
21
17
19
18
9
22
11
10
27
20
16
8
24
13
4
3
23
12
5
25
14
6
15
7
2
1
n:=Length(b);
inc:=List([1..Length(b)],x->Filtered([1..n],y->Length(b[y])>Length(b[x])
and RankMat(Concatenation(b[x],b[y]))=Length(b[y])));
maxinc:=List([1..n],x->Filtered(inc[x],y->not ForAny(inc[x],z->y in inc[z])));