I need to find all subgroups of $ G $= {1, -1, i, -i}, with $G \leq$ ($ \mathbb {C}$- {0}, $\cdot$)

Question

If you can guide me with this exercise, to be able to do other similar ones, it would be of great help to me.

Answer 1

I understand that you want to consider the group $G$ as the set $\{\pm 1, \pm i\}$ under regular complex multiplication. Let's see which groups each element generates by self-multiplication:

• $1 \mapsto 1 \implies \{1\}$
• $-1 \mapsto (-1)^2 = 1 \mapsto 1 \implies \{-1,1\}$
• $i \mapsto i^2 = -1 \mapsto -i \mapsto 1 \implies G$
• $-i \mapsto (-i)^2 = -1 \mapsto i \mapsto 1 \implies G$

So you have 2 subgroups: trivial one, full group, and of size 2.