Determine the number of elements in the direct product

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Reference no: EM131069943

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 Zm ⊕ Zn 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 Dn is generated by a rotation a of order n and a reflection b, that is Dn = (a, b).

Group

Element

Order

Z30 Z15

(13, 13)

          

Z30 Z15

(14, 13)

          

Z30 U (15)

(13, 13)

          

D10 U (15)

(a2b1, 13)

          

Reference no: EM131069943

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