Cayley Digraph

Please, can you graph this Cayley Digraph, like is in the attachment. I want you graph using Microsoft Word. Each dot is a vertex and the arrows are the edges. Please, do not confuse the vertex with the little 3 dots which means continuous the graph. If you have a question, let me know. I explain in attach you use two different kind of arrows, one indicate the horizontal edges and another indicates the vertical edges.

Groups (Prove)

I need help with the following question : For a fixed element a of a group G, the set Ca ={ x element-of G|ax=xa} is the centralizer of a in G. Prove that for any a element-of G, Ca is a subgroup of G.

Groups - For a fixed positive integer n

** decide whether each of the given sets is a group with respect to the indicated operation. 1- For a fixed positive integer n, the set of all complex numbers x such that x^n=1(that is, the set of all nth roots of 1),with operation multiplication. 2-The set of all complex numbers x that have absolute value1, with operation multiplication. Recall that the absolute value of a complex number x written in the form x= a + bi, with a and b real, is given by lxl=l a+bi l=Sqrta^2+b^2

Groups - Suppose G is a finite group of even order. Show that there is at least one element a in G, a (is not equal to) e, such that a=a^-1

Prove the following : Suppose G is a finite group of even order. Show that there is at least one element a in G, a (is not equal to) e, such that a=a^-1

Quadratic Applications - Bicycle Shop

The manager of a bicycle shop has found that, a a price (in dollars) of p(x) = 150 - x/4 per bicycle, x bicycles will be sold. a.find an expression for the total revenue from the sale of x bicycles. b.find the number of bicycle sales that leads to maximum revenue. c.find the maximum revenue.