If G = {1,-1,i,-i} is a multiplicative group then what is the order of -1?

Question

Answer

Order of -1 in the Multiplicative Group G

Definition of Order: In a multiplicative group, the order of an element is the smallest positive integer n such that a^n = e, where a is the element and e is the identity element (1 in this case).
Order of -1: To find the order of -1 in the group G = {1, -1, i, -i}, we need to determine the smallest positive integer n such that (-1)^n = 1.
Calculating Powers of -1:
- When n is odd, (-1)^n = -1.
- When n is even, (-1)^n = 1.
Conclusion: Since the order of -1 is the smallest positive integer n such that (-1)^n = 1, we can see that the order of -1 is 2 in the group G.