Final answer:
The orders of the elements {1, -1, i, -i} are 1, 2, 4, and 4 respectively, as determined by the smallest power each element must be raised to in order to reach the identity element of the group.
Step-by-step explanation:
The order of an element in a group is defined as the smallest positive integer n such that the element raised to the n-th power is equal to the identity element of the group. For the group {1, -1, i, -i}, we find the order of each element by raising them to successive powers until we get the identity element, which is 1 in this group.
- For 1: Any number raised to any power is itself, so the order is 1.
- For -1: Raising -1 to the power of 2 gives us 1, so the order is 2.
- For i: Raising i to the power of 4 gives us 1 (i^2 = -1 and (-1)^2 = 1), so the order is 4.
- For -i: Similarly, raising -i to the power of 4 gives us 1 (-i^2 = -(-1) = 1), so the order is 4.