### Determine the number of elements in the direct product

Assignment Help Mathematics
##### 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)

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