MATH 1101 Lecture 9: Math 1101 notes 5-1

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.

| Graph the function using a combination of transformations applied to the graph of a basic function g(x) = 5(x + 4)2-1 Tools Actions Clear A 1. Select an object from the Tools menu to the left Deiete 2. Enter coordinates in Object Properties below, or use the mouse to place and move objects. To enter a fractional r decimal coordinate, use No SolurionObject Properties l View our tutorial videos Select a Tool or Object to Begin Help WebAssign graphing tool Need Help? LReadIt?1LTalkto anteri