Exercises 1

All representations are over ℂ, unless the contrary is stated.

In Exercises 01–11 determine all 1-dimensional representations of the given group.

1 ∗ C_{2}  2 ∗∗ C_{3}  3 ∗∗ C_{n}  4 ∗∗ D_{2}  5 ∗∗ D_{3}
6 ∗∗∗ D_{n}  7 ∗∗∗ Q_{8}  8 ∗∗∗ A_{4}  9 ∗∗∗∗ A_{n}  10 ∗∗ ℤ

11 ∗∗∗∗ D_{∞} = ⟨r, s : s^{2} = 1, rsr = s⟩

Suppose G is a group; and suppose g, h ∈ G. The element [g, h] = ghg^{−1}h^{−1} is called the commutator of g and h. The subgroup G′ ≡ [G, G] is generated by all commutators in G is called the commutator subgroup, or derived group of G.

12 ∗∗∗ Show that G′ lies in the kernel of any 1-dimensional representation ρ of G, ie ρ(g) acts trivially if g ∈ G′.

13 ∗∗∗ Show that G′ is a normal subgroup of G, and that G/G′ is abelian. Show moreover that if K is a normal subgroup of G then G/K is abelian if and only if G′ ⊂ K. [In other words, G′ is the smallest normal subgroup such that G/G′ is abelian.)

14 ∗∗ Show that the 1-dimensional representations of G form an abelian group G^{∗} under multiplication. [Nb: this notation G^{∗} is normally only used when G is abelian.]

15 ∗∗ Show that C_{n}^{∗} ≅ C_{n}.

16 ∗∗∗ Show that for any 2 groups G, H

(G × H)^{∗} = G^{∗} × H^{∗}.

17 ∗∗∗∗ By using the Structure Theorem on Finite Abelian Groups (stating that each such group is expressible as a product of cyclic groups) or otherwise, show that

A^{∗} ≅ A

for any finite abelian group A.

18 ∗∗ Suppose Θ : G → H is a homomorphism of groups. Then each representation α of H defines a representation Θα of G.

19 ∗∗∗ Show that the 1-dimensional representations of G and of G/G′ are in oneone correspondence.

In Exercises 20–24 determine the derived group G′ of the given group G.

\begin{matrix}20 ∗∗∗ C_{n} & 21 ∗∗∗∗ D_{n} & 22 ∗∗ ℤ & 23 ∗∗∗∗ D_{∞} \\ 24 ∗∗∗ Q_{8} & 25 ∗∗∗ S_{n} & 26 ∗∗∗ A_{4} & 27 ∗∗∗∗ A_{n} \end{matrix}

Exercises 2

All representations are over ℂ, unless the contrary is stated. In Exercises 01–15 determine all 2-dimensional representations (up to equivalence) of the given group.

\begin{matrix}1 ∗∗ C_{2} & 2 ∗∗ C_{3} & 3 ∗∗ C_{n} & 4 ∗∗∗ D_{2} & 5 ∗∗∗ D_{4} \\ 6 ∗∗∗ D_{5} & 7 ∗∗∗∗ D_{n} & 8 ∗∗∗ S_{3} & 9 ∗∗∗∗ S_{4} & 10 ∗∗∗∗∗ S_{n} \\ 11 ∗∗∗∗ A_{4} & 12 ∗∗∗∗∗ A_{n} & 13 ∗∗∗ Q_{8} & 14 ∗∗ ℤ & 15 ∗∗∗∗ D_{∞} \end{matrix}

16 ∗∗∗ Show that a real matrix A ∈ Mat(n, ℝ) is diagonalisable over ℝ if and only if its minimal polynomial has distinct roots, all of which are real. 17 ∗∗∗ Show that a rational matrix A ∈ Mat(n, ℚ) is diagonalisable over ℚ if and only if its minimal polynomial has distinct roots, all of which are rational. 18 ∗∗∗∗ If 2 real matrices A, B ∈ Mat(n, ℝ) are similar over ℂ, are they necessarily similar over ℝ, ie can we find a matrix P ∈ GL(n, ℝ) such that B = P AP^{−1}? 19 ∗∗∗∗ If 2 rational matrices A, B ∈ Mat(n, ℚ) are similar over ℂ, are they necessarily similar over ℚ? 20 ∗∗∗∗∗ If 2 integral matrices A, B ∈ Mat(n, ℤ) are similar over ℂ, are they necessarily similar over ℤ, ie can we find an integral matrix P ∈ GL(n, ℤ) with integral inverse, such that B = P AP^{−1}? The matrix A ∈ Mat(n, k) is said to be semisimple if its minimal polynomial has distinct roots. It is said to be nilpotent if A^{r} = 0 for some r > 0. 21 ∗∗∗ Show that a matrix A ∈ Mat(n, k) cannot be both semisimple and nilpotent, unless A = 0. 22 ∗∗∗ Show that a polynomial p(x) has distinct roots if and only if

gcd (p(x), p′(x)) = 1.

23 ∗∗∗∗ Show that every matrix A ∈ Mat(n, ℂ) is uniquely expressible in the form

A = S + N,

where S is semisimple, N is nilpotent, and

SN = NS.

(We call S and N the semisimple and nilpotent parts of A.) 24 ∗∗∗∗ Show that S and N are expressible as polynomials in A. 25 ∗∗∗∗ Suppose the matrix B ∈ Mat(n, ℂ) commutes with all matrices that commute with A, ie

AX = XA ⟹ BX = XB.

Show that B is expressible as a polynomial in A.

Exercises 3

In Exercises 01–10 determine all simple representations of the given group over ℂ.

\begin{matrix}1 ∗∗ C_{2} & 2 ∗∗ C_{3} & 3 ∗∗ C_{n} & 4 ∗∗∗ D_{2} & 5 ∗∗∗ D_{4} \\ 6 ∗∗∗ D_{5} & 7 ∗∗∗∗ D_{n} & 8 ∗∗∗ S_{3} & 9 ∗∗∗∗ A_{4} & 10 ∗∗∗∗∗ Q_{8} \end{matrix}

In Exercises 11–20 determine all simple representations of the given group over ℝ.

\begin{matrix}11 ∗∗ C_{2} & 12 ∗∗∗ C_{3} & 13 ∗∗∗ C_{n} & 14 ∗∗∗ D_{2} & 15 ∗∗∗∗ D_{4} \\ 16 ∗∗∗∗ D_{5} & 17 ∗∗∗∗ D_{n} & 18 ∗∗∗∗ S_{3} & 19 ∗∗∗∗∗ A_{4} & 20 ∗∗∗∗∗ Q_{8} \end{matrix}

In Exercises 21–25 determine all simple representations of the given group over the rationalsℚ.

21 ∗∗∗∗ C_{n} 22 ∗∗∗∗ D_{n}  23 ∗∗∗∗ S_{3}  24 ∗∗∗ Q_{8}  25 ∗∗∗∗∗ A_{4}