Part A-

1. Establish or each of the following statements as being true false. Justify each answer fully.

(a) Z₅ ⊕ Z₁₂ ≅ Z₆₀

(b) Z₅ ⊕ Z₁₀ ≅ Z₅₀

(c) Z₁₂ ⊕ Z₄ ≅ Z₂₄ ⊕ Z₂

(d) U(180) ≅ U(112)

2. (a) Express U(77) as an external direct product of groups of the form Zn in three different ways.

(b) Express Aut(Z₅₅) as Z_m ⊕ Z_n for some m and n.

3. Suppose φ is an isomorphism from Z₅ ⊕ Z₁₁ to Z₅₅, and φ(2, 3) = 4. Find the element that φ maps to 1.

4. Give an example of an infinite non-Abelian group with precisely six elements of finite order.

5. Find two (distinct) subgroups of order 30 in Z₅₀ ⊕ Z₆₀.

6. Determine the number of elements of order 10 and the number of cyclic subgroups of order 10 in Z₂₀ ⊕ Z₁₅.

Part B-

1. Determine the order of each of the following elements in the respective products of groups (D₃₀ denotes the dihedral group of order 60 which is generated by a, b where b is a reflection and a is a rotation).

2. For each of the following pairs of groups G1, G2, determine the number of elements in the direct product G1 ⊕G2 of the given order (D₃ is the dihedral group of order 6).

3. Find the order of each of the following elements n the respective groups:

Note:  In the table below, the dihedral group D_n is generated by a rotation a of order n and a reflection b, that is D_n = (a, b).