For each of the following pairs of groups g1 g2 determine

Question

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).

element	product	order
(23, 9)	Z30 ⊕ Z22
(23, 9)	Z30 ⊕ U(22)
(19, a15)	U(30) ⊕ Z22

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).

G1	G2	k	Number of elements in G1 ⊕G2 of order k
Z6	Z12	4	---------
U(10)	U(13)	4	---------
D3	Q8	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).

Group	Element	Order
Z30 ⊕ Z15	(13, 13)
Z30 ⊕ Z15	(14, 13)
Z30 ⊕U (15)	(13, 13)
D10 ⊕U (15)	(a2b1, 13)