1. Notation #

a) Notate each move in the following video. You should be able to put your solution into the Moves section on the right side of alg.cubing.net, then see the same result after pressing the play button on the left side.

b) Invert the move sequence R U R' U' R' F R F'.

Hint: If you start with a solved cube, do this sequence of moves, then its inverse, the cube should be solved at the end. You can check your answer on https://alg.cubing.net/.

2. Group Definition. #

Your tutor Manu tells his friend Zod about a new group which he came up with. He says his group is $(\mathbb{Z}^+, \cdot).$ The group elements are the positive integers (1, 2, 3, ...), and the group operation is multiplication. Zod says Manu is wrong, and his group is not a valid group. Who is right and why?

3. Group Properties. #

Show the following statement is true: Suppose $G$ is a group with elements $a, b, x$ and binary operator $*.$ If $a * x = b * x,$ then $a=b$.

Hint: This problem has the form “If <assumption>, then <result>.” Start with the assumption, then use group properties to show the result. Section 2.3.1 and the inverse property of groups might be helpful here.