Question

1. Check whether the set S=R - {-1} is a group under the binary operation ‘*’defined as for any two elements .
2. i. State the relation between the order of a group and the number of binary operations that can be defined on that set.
ii. How many binary operations can be defined on a set with 4 elements?
3. Discuss the group theory concept behind the Rubik’s cube.

Answer 1

2(i) There are n elements in the set and binary operation i.e 2 operations can be applied on each of them in relation hence the total number of combinations will be $2^n . 2^{n-1} ... 2^2 . 2 = 2^{1+2+...n} = 2^{\frac{n(n-1)}{2}}$ .

(ii) i. How many binary operations can be defined on a set with 4 elements? 

The formula is $2^{n(n-1)/2}$ where n is number of elements as explained in 2(i).

Hence So, Binary operations can be defined on a set with 4 elements $= 2^{4(4-1)/2} = 2^6 = 64$ .

3) On the Rubik's Cube, there are 54 facets that can be arranged and rearranged through

twisting and turning the faces. Any position of the cube can be describe as a permutation from the solved state. Thus, the Rubik's Cube group is a subgroup of a permutation

group of 54 elements.