I have a few problems I can't figure out, I'm sure they're not too complex, I just can't grasp how I'm supposed to tackle these.... (* means binary operation in these)

1. Let (G, *) be a group and let g be some fixed element of G. Show that G = {g*x such that x is in G}

2. Let G be a nonempty set and let * be an associative binary operation on G. Assume that both the left and right cancellation laws hold in (G, *). Assume moreover that G is finite. Show that (G, *) is a group. (Obviously, * being associative is a part of what makes it a group. However, I don't know how to show how an identity exists and an inverse. I don't get how the group being finite has anything to do with this.)

3. Let G be a group and let x in G be an element of order 18. Find the orders of x^2, x^3, x^4, x^5, x^12.

4. Show that if G is a finite group, then every element of G is of finite order.

5. Give an example of an infinite group G such that every element of G has finite order.