Question

Prove that the set of complex numbers {1,−1,i,−i} under

multiplication operation is a cyclic group.Find the generators of

cycle

Answer 1

The group that contains a finite number of elements is cyclic, in case there is an element $g$ of this group such that all elements have the form: $g^j$, where $j\in{\mathbb{N}}$. I.e., all elements can be obtained from $g$. In case of set $\{1,-1,i,-i\}$ we have: $i^2=-1,$ $i^3=-i$, $i^4=1$. Thus, all elements can be recovered from $i$. The group is cyclic. $i$ is a generator of the group. Consider other elements of the group: $(-i)^2=-1$, $(-i)^3=i$, $(-i)^4=1$. . $(-1)^2=1$, $1^2=1.$

Thus, the answer is: the group is cyclic, the generators are $(-i)$ and $i$. $(-1)$ generates a subgroup: $\{-1,1\}$. $1$ generates only itself.