Reference no: EM131069943
Part A
1. Establish or each of the following statements as being true false. Justify each answer fully.
(a) Z_{5} ⊕ Z_{12} ≅ Z_{60}
(b) Z_{5} ⊕ Z_{10} ≅ Z_{50}
(c) Z_{12} ⊕ Z_{4} ≅ Z_{24} ⊕ Z_{2}
(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(Z_{55}) as Z_{m} ⊕ Z_{n} for some m and n.
3. Suppose φ is an isomorphism from Z_{5} ⊕ Z_{11} to Z_{55}, and φ(2, 3) = 4. Find the element that φ maps to 1.
4. Give an example of an infinite nonAbelian group with precisely six elements of finite order.
5. Find two (distinct) subgroups of order 30 in Z_{50} ⊕ Z_{60}.
6. Determine the number of elements of order 10 and the number of cyclic subgroups of order 10 in Z_{20} ⊕ Z_{15}.
Part B
1. Determine the order of each of the following elements in the respective products of groups (D_{30} 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)

Z_{30} ⊕ Z_{22}


(23, 9)

Z_{30} ⊕ U(22)


(19, a^{15})

U(30) ⊕ Z_{22}


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 (D_{3} is the dihedral group of order 6).
G_{1}

G_{2}

k

Number of elements in G_{1} ⊕G_{2} of order k

Z_{6}

Z_{12}

4



U(10)

U(13)

4



D_{3}

Q_{8}

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)

