mth3003 predicted paper

MTH3003 Group Theory — Practice Exam

Instructions: Answer ALL questions. Show all working. This exam covers the material from Weeks 1–7 of the module.

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Question 1 — Permutation Computation [Sections 1a–1d]

Let $g=(1638)(257)$ and $h=(142)(3756)$ be elements of $S_9$.

(a) Write $g$ out as a full two-row mapping (i.e. state where each element $1,2,\dots ,9$ is sent). [2 marks]

(b) Compute $g^{-1}$. [2 marks]

(c) Compute $gh$ as a product of disjoint cycles. [4 marks]

(d) Find $o(g)$ and $o(h)$. [2 marks]

(e) Find an element of $S_{12}$ with order 20, or explain why no such element exists. [3 marks]

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Question 2 — Solving a Permutation Equation [Section 1e]

Find $\sigma \in S_8$ satisfying

$$(145)(2378)=(162)\sigma (354).$$

Verify your answer by substituting back. [6 marks]

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Question 3 — Subgroups and Cyclic Groups [Sections 2a, 2c, 2d]

(a) Let $G$ be a group with subgroups $H\le G$ and $K\le G$. Prove that $H\cap K\le G$. [4 marks]

(b) Let $g=(1254)(36)\in S_7$. Find $o(g)$, list all elements of $⟨g⟩$, and list all elements of $⟨g^2⟩$. [5 marks]

(c) Let $p$ be a prime number and $|G|=p$. Prove that $G$ is cyclic. [4 marks]

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Question 4 — Dihedral Groups [Section 4]

Consider the dihedral group $D_{12}$ (symmetries of a regular hexagon), with vertices labelled $1,2,3,4,5,6$ anticlockwise. Let $\rho$ denote anticlockwise rotation by $\pi /3$ and $\sigma$ the reflection through the axis passing through vertex 1.

(a) Write $\rho$ and $\sigma$ in cycle notation. [2 marks]

(b) Write $\sigma {\rho }^2$ and $\sigma {\rho }^4$ in cycle notation. [4 marks]

(c) Simplify ${\rho }^4\sigma {\rho }^2\sigma$ to the form ${\rho }^i$ or $\sigma {\rho }^i$. [4 marks]

(d) Using the relation $\rho \sigma =\sigma {\rho }^{-1}$, prove that $⟨\rho ,\sigma ⟩=D_{12}$. [4 marks]

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Question 5 — Normal Subgroups and Lagrange’s Theorem [Sections 2b, 3]

(a) State Lagrange’s Theorem. [2 marks]

(b) Prove that $S_7$ has no subgroup of order 11. [2 marks]

(c) Let $G$ be a group, $H\le G$, and $N⊴G$. Prove that $H\cap N⊴H$. [5 marks]

(d) Let $H\le G$ and $N⊴G$. Using the Order Switching Lemma, prove that $HN\le G$. [6 marks]

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Question 6 — Homomorphisms and Isomorphisms [Section 5]

(a) Define what it means for a map $\theta :G\to H$ to be a group homomorphism. [1 mark]

(b) Let $\theta :G\to H$ be a homomorphism.

• (i) Prove $\theta (e_G)=e_H$. [2 marks]
• (ii) Prove $\text{Ker}(\theta )⊴G$. [4 marks]

(c) Consider the map $\varphi :C_5\to C_5$ defined by $\varphi (g)=g^3$. Determine whether $\varphi$ is a homomorphism, and if so, whether it is an isomorphism. Justify your answers fully. [4 marks]

(d) Prove that $D_6\cong S_3$ by constructing an explicit isomorphism. [5 marks]

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Question 7 — Isomorphism Theorems and Quotient Groups [Sections 6, 10]

Let $\det :{\text{GL}}_2(\mathbb{R})\to {\mathbb{R}}^{\ast }$ denote the determinant map, which is a surjective homomorphism.

(a) State the First Isomorphism Theorem. [2 marks]

(b) Prove that ${\text{SL}}_2(\mathbb{R})⊴{\text{GL}}_2(\mathbb{R})$. [3 marks]

(c) Using the First Isomorphism Theorem, show that ${\text{GL}}_2(\mathbb{R})/{\text{SL}}_2(\mathbb{R})\cong ({\mathbb{R}}^{\ast },\times )$. [3 marks]

(d) Let $P=\{M\in {\text{GL}}_2(\mathbb{R}):\det (M)>0\}$. Using the Third Isomorphism Theorem, show that

$$({\text{GL}}_2(\mathbb{R})/{\text{SL}}_2(\mathbb{R}))/(P/{\text{SL}}_2(\mathbb{R}))\cong {\text{GL}}_2(\mathbb{R})/P.$$

Hence identify ${\text{GL}}_2(\mathbb{R})/P$ as a familiar group. [4 marks]

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Question 8 — The Alternating Group and Signature Function [Section 7]

(a) Let $g=(14628)(397)(510)$ and $h=(13)(25)(46)(78)(910)$ be elements of $S_{10}$.

• (i) Compute $\sigma (g)$ and state whether $g\in A_{10}$. [2 marks]
• (ii) Compute $\sigma (h)$ and state whether $h\in A_{10}$. [2 marks]
• (iii) Compute $\sigma (g^{-1}hg)$ and state whether $g^{-1}hg\in A_{10}$. [2 marks]

(b) List all cycle shapes of elements in $S_4$, determine which correspond to even permutations, and hence find $|A_4|$. [5 marks]

(c) Using the signature function $\sigma :S_n\to C_2$, prove that $A_n⊴S_n$ and $S_n/A_n\cong C_2$. [4 marks]