The multiplicative group {1, -1, i, -i} is a cyclic group, its generators are

Question

Question

This question was previously asked in

UP TGT Mathematics 2021 Official Paper

Answer 1

Given:

G = {1, -1, i, -i} is a cyclic group of order 4.

⇒ o(a) = o(G)

Evidently;

⇒ i = i, i²= -1, i³ = -i, i⁴ = 1

Here identity element e of G is 1.

Thus, i4 = 1and -1⁴= 1 i.e i^r ≠ 1 for Any r < 1.

This → o(i) = 4 = o(G)

⇒ i and -i are the generator of G.

Download Solution PDF