1. **Problem:** Discuss why any group is isomorphic to a group of permutations. **Explanation:** This is Cayley's theorem. **Step 1:** Let $G$ be any group. Consider the set $G$ itself. **Step 2:** Define a function $\varphi : G \to S_G$ where $S_G$ is the group of all permutations on $G$. **Step 3:** For each $g \in G$, define $\varphi(g)$ as the permutation of $G$ given by left multiplication: $\varphi(g)(x) = gx$ for all $x \in G$. **Step 4:** Show $\varphi$ is a group homomorphism: $\varphi(gh)(x) = (gh)x = g(hx) = \varphi(g)(\varphi(h)(x))$. **Step 5:** $\varphi$ is injective because if $\varphi(g) = \varphi(h)$, then $gx = hx$ for all $x$, so $g = h$. **Conclusion:** $G$ is isomorphic to the subgroup $\varphi(G)$ of $S_G$, a group of permutations. 2. **Problem:** State and prove the second isomorphism theorem of groups. **Statement:** Let $H$ be a subgroup and $N$ a normal subgroup of $G$. Then $HN$ is a subgroup of $G$, $N$ is normal in $HN$, and $$H/(H \cap N) \cong HN/N.$$ **Proof:** 1. $HN = \{ hn \mid h \in H, n \in N \}$ is a subgroup. 2. $N$ is normal in $HN$ because $N$ is normal in $G$. 3. Define $\varphi : H \to HN/N$ by $\varphi(h) = hN$. 4. $\varphi$ is a homomorphism with kernel $H \cap N$. 5. By the first isomorphism theorem, $H/(H \cap N) \cong \varphi(H) = HN/N$. 3. **Problem:** Prove that a group $G$ is abelian if and only if $(ab)^2 = a^2 b^2$ for all $a,b \in G$. **Proof:** 1. If $G$ is abelian, then $ab = ba$, so $(ab)^2 = abab = aabb = a^2 b^2$. 2. Conversely, assume $(ab)^2 = a^2 b^2$ for all $a,b$. 3. Expand: $(ab)^2 = abab = a^2 b^2$ implies $abab = aabb$. 4. Multiply both sides on the left by $a^{-1}$ and on the right by $b^{-1}$: $$a^{-1} a b a b b^{-1} = a^{-1} a a b b b^{-1} \implies b a = a b,$$ so $G$ is abelian. 4. **Problem:** If $K$ is a normal subgroup of $G_1$ and $f : G \to G_1$ is a homomorphism, prove that $f^{-1}(K)$ is a normal subgroup of $G$ containing $\ker f$. **Proof:** 1. $f^{-1}(K) = \{ g \in G \mid f(g) \in K \}$. 2. Since $K$ is a subgroup, $f^{-1}(K)$ is a subgroup. 3. For normality, for any $g \in G$ and $x \in f^{-1}(K)$, $$f(g x g^{-1}) = f(g) f(x) f(g)^{-1} \in K,$$ because $K$ is normal in $G_1$. 4. Hence $g x g^{-1} \in f^{-1}(K)$, so $f^{-1}(K)$ is normal. 5. Also, $\ker f = f^{-1}(\{e\}) \subseteq f^{-1}(K)$. 5. **Problem:** Define and give examples of infinite cyclic group and finite simple group. **Definitions:** - Infinite cyclic group: A group generated by a single element of infinite order, isomorphic to $(\mathbb{Z}, +)$. - Example: $(\mathbb{Z}, +)$ itself. - Finite simple group: A nontrivial finite group with no normal subgroups other than the trivial group and itself. - Example: The alternating group $A_5$. 6. **Problem:** Let $G_1, G_2$ be groups. (i) Prove $G = G_1 \times G_2$ is a group. **Proof:** 1. Define operation componentwise: $(g_1, g_2)(h_1, h_2) = (g_1 h_1, g_2 h_2)$. 2. Associativity follows from associativity in $G_1$ and $G_2$. 3. Identity is $(e_1, e_2)$. 4. Inverse of $(g_1, g_2)$ is $(g_1^{-1}, g_2^{-1})$. (ii) Find normal subgroups $H, K$ such that $G = HK$, $H \cap K = \{e\}$. 1. Let $H = G_1 \times \{e_2\}$ and $K = \{e_1\} \times G_2$. 2. Both are normal in $G$. 3. $HK = G$ and $H \cap K = \{(e_1, e_2)\}$. 7. **Problem:** Explain and give example of internal direct product of normal subgroups. **Explanation:** A group $G$ is an internal direct product of normal subgroups $H_i$ if each $H_i$ is normal, $G = \prod H_i$, and $H_i \cap \prod_{j \neq i} H_j = \{e\}$. **Example:** $\mathbb{Z}_6 \cong \mathbb{Z}_2 \times \mathbb{Z}_3$. 8. **Problem:** Let $N$ be a normal subgroup of $G$. Define relation $\theta_N$ on $G$ by $(x,y) \in \theta_N \iff xy^{-1} \in N$. (1) Prove $\theta_N$ is a congruence relation. (2) Show $[e]_{\theta_N} = N$. (3) Show $[a]_{\theta_N} = a [e]_{\theta_N}$ for all $a \in G$. (4) Clarify for $G = \{1, -1, i, -i\}$. **Proof:** 1. $\theta_N$ is an equivalence relation because $N$ is normal. 2. $[e]_{\theta_N} = \{ x \in G \mid x e^{-1} = x \in N \} = N$. 3. $[a]_{\theta_N} = \{ x \mid x a^{-1} \in N \} = a N = a [e]_{\theta_N}$. 4. For $G = \{1, -1, i, -i\}$, $N$ could be $\{1, -1\}$, and cosets partition $G$ accordingly. 9. **Problem:** Construct an epimorphism $f : S_3 \to G = \{1, -1\}$. **Construction:** 1. Define $f(\sigma) = 1$ if $\sigma$ is even permutation, $-1$ if odd. 2. $f$ is surjective homomorphism. 3. The congruence $\theta_f$ on $S_3$ is kernel equivalence: $(x,y) \in \theta_f$ if $f(x) = f(y)$. 4. The canonical homomorphism $\gamma : S_3 \to S_3 / \theta_f$ maps $x$ to its coset modulo kernel. **Final answers:** - Cayley's theorem: Any group is isomorphic to a permutation group. - Second isomorphism theorem: $H/(H \cap N) \cong HN/N$. - $G$ abelian iff $(ab)^2 = a^2 b^2$. - $f^{-1}(K)$ normal in $G$ containing $\ker f$. - Infinite cyclic group example: $\mathbb{Z}$; finite simple group example: $A_5$. - $G_1 \times G_2$ is a group; $H = G_1 \times \{e\}$, $K = \{e\} \times G_2$ normal with $G = HK$. - Internal direct product explained with example. - $\theta_N$ is congruence relation with cosets. - Epimorphism $f : S_3 \to \{1,-1\}$ given by sign map.