Appendix C — Homework 2

Exercise C.1 Let \(G\) be a group and take \(h, k \in G\). The (group-wise) commutator of \(h\) and \(k\) is given by \[ [h,k] \coloneqq h^{-1} k^{-1} h k. \] Notice that \([h,k]=e\) if and only if \(hk = kh\), so in particular the commutator in an abelian group is always trivial. One might say that commutators, taken all together, measure how badly a group fails to be abelian.

  1. Show that \(g[h,k]g^{-1}\) can be written as a commutator.1 Conclude that the subgroup generated by all commutators in \(G\), known as the commutator subgroup and written \([G,G]\), is normal in \(G\).

  2. The abelianization of \(G\) is defined as the quotient \({G}^{\mathrm{Ab}} \coloneqq G/[G,G]\). Show that \({G}^{\mathrm{Ab}}\) is abelian.2

  3. Compute the abelianization of \(\mathcal{A}_{4} = \langle (1\ 2\ 3), (1\ 2)(3\ 4) \rangle\).

  4. Let \(\pi: G \twoheadrightarrow{G}^{\mathrm{Ab}}\) be the canonical homomorphism. If \(\varphi: G \to A\) is a group homomorphism and \(A\) is abelian, show that there is a homomorphism \(\tilde{\varphi}: {G}^{\mathrm{Ab}} \to A\) such that \(\varphi = \tilde{\varphi} \circ \pi\). This is the universal property of abelianizations.

Exercise C.2 Let \(G\) be a finite group and write \(\rho:G \to \operatorname{GL}(\mathbb{C}[G]) = \operatorname{GL}_{|G|}(\mathbb{C})\) for the regular representation (Example 4.8). Show that the set \[ \operatorname{im}(\rho) = \left\{ \rho(g) \mid g \in G \right\} \subset \operatorname{Mat}_{|G|}(\mathbb{C}) \] is linearly independent.

Exercise C.3 Let \(G\) be finite, and let \(N \unlhd G\). For any \(\rho: G \to \operatorname{GL}(V)\), define the \(N\)-invariants of \(V\) (cf. Example 4.15) as \[ V^N \coloneqq \left\{ x \in V \mid \rho(n) x = x \text{ for all } n \in N \right\}. \]

  1. Prove that \(V^N\) is a \(G\)-subrepresentation of \(V\).

  2. Does \(V^N\) need to be trivial as a \(G\)-representation, i.e., must \(g \cdot x = x\) for all \(g \in G\) and \(x \in V^N\)? Prove or disprove.

Exercise C.4 Let \(G\) be a finite group with \(\rho: G \to \operatorname{GL}(V)\).

  1. If \(\rho\) has degree \(2\) or \(3\), show that \(\rho\) is irreducible if and only if there is no common eigenvector for every \(A \in \operatorname{im}\rho\).

    Hint: \(3 = 2 + 1\) and \(2 = 1 + 1\).

  2. Given an example to show the claim is false if the degree of \(\rho\) is \(4\).

Exercise C.5  

  1. Prove that there is a representation \(\rho: \mathcal{Q}_{8}\to \operatorname{GL}_3(\mathbb{C})\) with \[ \rho(i) = \frac{1}{2} \begin{pmatrix} 1-i & 0 & -1-i \\ 0 & 2i & 0 \\ -1-i & 0 & 1-i \end{pmatrix} \quad \text{ and } \quad \rho(j) = \frac{1}{2} \begin{pmatrix} -1 & \sqrt{2} & 1 \\ -\sqrt{2} & 0 & -\sqrt{2} \\ 1 & \sqrt{2} & -1 \end{pmatrix} \]

    Hint: What relations do you need to check?

  2. Decompose \(\rho\) into irreducible representations.

Exercise C.6 Here we work towards classifying the finite subgroups of \(\operatorname{SO}(3)\), i.e., real representations of degree \(3\). Towards that end, let \(G \leq \operatorname{SO}(3)\) be finite and nontrivial, and write \(S^2 \subset \mathbb{R}^3\) for the unit sphere. Note that \(\operatorname{SO}(3) ⟳S^2\) by left multiplication, since \[ \| Rx \|^2 = \langle Rx, Rx \rangle = \langle x, x \rangle = \| x \|^2 = 1 \] for every \(R \in \operatorname{SO}(3)\) and \(x \in S^2\). Thus, via inclusion, there is a natural action \(G ⟳S^2\).

  1. Let \(R \in \operatorname{SO}(3)\). Show that \(R\) has \(1\) as an eigenvalue.

    Hint: Consider \(\det(R - \mathbb{I})\) and recall that \((-1)^3 = -1\).

  2. If \(R \not = \mathbb{I}\), show that \(\operatorname{Fix}(R) \coloneqq \{ x \in S^2 \mid Rx = x \}\) consists of two antipodal points: \(\operatorname{Fix}(R) = \{y, -y\}\).

  3. Let \(Y \subset S^2\) be the set of elements fixed by at least one non-trivial \(g \in G\): \[ Y \coloneqq \bigcup_{g \in G \setminus \{\mathbb{I}\}} \operatorname{Fix}(g) \] Show that \(G ⟳Y\).

  4. Decomposing \(G ⟳Y\) into distinct orbits \(O_1, \dots, O_r\), use the Orbit-Stabilizer Theorem to show \[ 2 - \frac{2}{|G|} = \sum_{i=1}^r \left( 1 - \frac{1}{k_i} \right), \] where \(k_i = \frac{|G|}{|O_i|}\).

  5. (Bonus) We must have \(r \leq 3\) (why?) and we know \(2 \leq k_i\) divides \(|G|\) (Lagrange’s Theorem). These equations put strong restrictions on the possible groups \(G\)! What solutions can you find?

  1. A possible approach is using that, for any fixed \(g \in G\), the map \(\psi_g: G \to G\) given by \(\psi_g(h) = ghg^{-1}\) is a homomorphism (indeed, an isomorphism).↩︎

  2. In this sense, \({G}^{\mathrm{Ab}}\) is the largest abelian quotient of \(G\), since we obtained it by collapsing the smallest subgroup containing all commutators.↩︎