5. (14 points) Suppose that : G→ H is a group homomorphism, K <H is a subgroup, and ¹(K) = {g €G | (g) € K}. Prove that -¹(K) is a subgroup of G. 6. (14 points) Suppose A, B4G are normal subgroups such that AnB = {e}. Prove that if a EA and be B, then ab = ba. IMPORTANT: You should not simply quote the Proposition from class stating that AB Ax B. Instead, you should supply the first part of that proof of that proposition, stating that the elements of A commute with the elements of B.

Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp. Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or (g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply Proposition 3.7.1.

Exercise 3 In this exercise you will show that multiplication of cosets in Z[i]/(5) is well-defined.

Given AC Z and x Є Z, we say that x is A-mirrored if and only if −x € A. We also define: MA= {x EZ: x is A-mirrored}.

Let A and B be finite sets. (a) Find and example of A and B for which |A U B|=

3.2.11 For the following proof sketch, determine the result that is being proved, and then turn the sketch into a formal proof.

Exercise 3.6.4 Suppose that G₁, G₂ are groups, N₁ ◄G₁, N₂ ◄ G₂ are normal subgroups. Prove that N₁ × N₂ ◄ G₁ x G₂, and (G₁ × G₂)/(N₁ × N₂) ≈G₁/N₁ × G₂/N₂. Hint: Theorem 3.6.5.