14. Ira has $100 in his savings account. He is looking at two savings plans. Under plan A, he will increase his account balance by $10 a year. Under plan B, he will increase his account balance by 10% each year.

     a. Write an equation for the amount Ira will have with plan A.

     b. Write an equation for the amount Ira will have with plan B.

     c. To the nearest hundredth, how much would Ira save in 3 years under plan A? How much would he save under plan B?

2. Let G be a group and a be a specific element of G. Define (a) If G is any abelian group, what can you say about Ca (for any a E G)? Explain (b) Prove that Ca is a subgroup of G (whether or not G is abelian) 3. Consider the group G = ZeXZ10, which contains 60 elemients all of the form (a,b) where a € 26 and b E Z1 Note that this is an additive group (a) What is the order of each of the following elements: (3,5), (2,5), 1), and (5,9)? (b) True or false: if an element g E (a) (i.e., g is in the subgroup generated by a), then (g) is a subgroup of (a) Answer this question at least for this particular group G, but if possible, in general. Justify your answer (c) Is G cyclic? What orders of elements in G are possible? Explain

3. Consider the group G Z xZo, which contains 60 elements all of the form (a, b) where a E Z6 and b e Zo Note that this is an additive group (a) What is the order of each of the following elements: (3,5), (2.5), (1,1), and (5,9)? (b) True or false: if an element g e(a)(ie., g is in the subgroup generated by a), then (g) is a subgroup of (a). Answer this question at least for this particular group G, but if possible, in general. Justify your answer (c) Is G cyclic? What orders of elements in G are possible? Explain 4. Consider the following groups of order 6: 26,22 × 23 and D3 Here are their group tables Z6 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 10 ,0 (1,2) (0,0 (0 (0,2) (a) For each of these three groups, find a set of generators (of smallest possible size) and defining equations. Say what each element is in terms of the generator(s) (b) For each of these three groups, draw a Cayley diagram. Label the vertices with their corresponding elements.