MATH 3175 Quiz: MATH 3175 Quiz 2.1 Fall 2010

5. Let f: R→ R be defined by f(x)= 2-3. Prove that f is a bijection. 6. Let G be a group and let a E G. Prove that the function f : G → G defined by f(x) = a * x is a bijection (Note that if G is a finite group this gives an alternate proof of the fact that each element of G appears precisely once in each row of the group table of G.)

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.

Q Search 8. Let r(x) f((h(x), where h(1)2, g(2) 3, h'(1) 4, g'(2) 5, and f'(3) 6. What is the value of r'(1)? (A) 1 (B) 3 (C) 20 (D) 120 (E) 720 mjl TEST 8. Do as indicated. Show your complete solutions. SCORING. CORRECT: 10 pts each. INCORRECT: 9 points. MAX SCORE: 50 points dy,f-10%(sec-1 5x8) 1. Find dxcot 1086