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 non-Abelian 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}.

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