Hi can someone answer me the three question please 1 Conside

Hi can someone answer me the three question please

1. Consider the set {1, i ,-1,-i} under our usual multiplication. Does this form a group? Complete an Operation Table.

2. describe the symmetry group of of the figure:

3. Describe the symmetry group of the figure:

Solution

1) The given set of elements formed a closed group under the operation (multiplication)

The first set forms a group since there exists an identity element (1), there exists inverse of each and every element

Inverse of 1 = 1

Inverse of -1 = -1

Inverse of i = -i

Inverse of -i = i

All the result of aXb lies in the group, hence the closure property is satisfied

associativity is satisfied i.e. aX(bXc) = (aXb)Xc

2) The given group has three tentacles structure and is a planar group, hence the group is C3

Symmetry in Cn group is across 2*pi/n radians which is equal to 2pi/3 radians or 120 degree

3) The given group has six tentacles structure and is a planar group, hence the group is C6

Symmetry in Cn group is across 2*pi/n radians which is equal to 2pi/6 radians or 60 degree