# Objects with a binary operator

1.
A group is a set of objects with a binary operator which has satisfy various conditions. Suppose G is a set of elements with a binary operation ⋆. Select each condition which is required for ( G , ⋆ ) to be an group:

For any elements f, g, and h in G, (f * g) * h = f * (g * h).

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There exist an identity element e in G such that for all g in G, g * e = e * g = g.

For all elements g in G, there exists an element g-1 in G such that g * g-1 = g-1 * g = e.

For any elements f, g in G, f * g = g * f.

For any elements f, g in G, the element f * g is also in G.

2.
What is the order of the following cycle given in cycle notation (Please enter an integer):

(1 2 3 4 5) (6 7 8)

3.
True or False: The following cycle has an even parity:

(1 2 3 4)

4.
True or False: The cube always has an even parity (the number of cubies exchanged from the starting position is always even).

5.
Find the order of the subgroup of the Rubik’s cube generated by the following element:

(R’ L F R L’ U’)

6.
Which of the following choices is the inverse of the following moves:

R U L R’

R’ U’ L’ R

R U L R

R L’ U’ R’

R’ L’ U’ R’

7.

True or False: The following two group elements are the same:
1. U D U
2. U2 D
3. Which of these choices is a valid cycle decomposition of the following permutation?
8
What is the corner/edge parity of the cube after applying the following move to the solved cube? R

Corners are even, edges are even.

Corners are odd, edges are odd.

Corners are even, edges are odd.

Corners are odd, edges are even.

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