#### Abstract Algebra

3. (14 points) Prove that (Z,+), the integers with addition, forms a group (you may assume "standard" facts about addition of integers). 4. (14 points) Draw the subgroup lattice of Z100-

Exercise 6. INVERSE TRANSFORM Determine the signal x(t) by computing the inverse Fourier transform of the following X(jw).

Exercise 3.6.6

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.

Exercise 3.7.2 Suppose G is a finite group, H, KG are normal subgroups, ged(|H|, |K|) = 1, and |G| = |H||K|. Prove that G H x K.

Exercise3.7.6 Prove that for all n≥2，we have Sn≌An为Z2,where Sn is the symmetric group and Anthe alternating subgroup (consisting of the even permutations).Hint:use Proposition 3.7.7.You have to find asubgroup K Sn with KZ2 (c.f.Erample 3.7.8).

Exercise 3.3.8

Exercise 2 Let G be a group of order 3²-5²-13 2925.

Exercise 2 Over the course of the parts of this exercise you will show that multiplication of cosets in Z[i]/Z is not well-defined. (a) Let a, a, b, VEZ. Prove that a +i and a' + i represent the same coset in Z[i]/Z; that is, prove that (a + i) + Z= (a' + i) + Z (b) Prove that band & represent the same coset in Z[i]/Z; that is, prove that b+z=b'+Z (c) Determine the conditions under which (a +i)b and (a' + i)b' represent the same coset in Z[i]/Z. That is, determine what additionally must be assumed in order to ensure that (a + i)b+Z= (a' + i)b' +Z. Be sure to prove your condition is the correct one. (d) Find specific values of a, a, b, b' € Z for which (a+i)b+Z‡ (a¹ + i)b +Z.

Decide whether each of the statements below is true or false. Circle your choice.