Determine the number of elements

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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.

Reference no: EM131069898

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