Understanding Group Theory Concepts

What is the order of an element in a group?

• The largest negative integer n such that x^n equals the identity element

• The smallest negative integer n such that x^n equals the identity element

• The smallest positive integer n such that x^n equals the identity element

• The largest positive integer n such that x^n equals the identity element

In the group of nonzero real numbers under multiplication, what is the order of -1?

• 3

• 1

• 2

• Infinite

Which of the following is a root of unity in the group of nonzero complex numbers?

• 2

• i

• 0

• -1

What is the condition for a 2x2 real matrix to have an inverse?

• It must be a diagonal matrix

• It must have a non-zero determinant

• It must have a zero determinant

• It must be a symmetric matrix

What is the order of the rotation matrix M given in the example?

• 12

• 10

• 8

• 6