The set $\{1,-1,i,-i\}$ consists of all 4th roots of unity. Under multiplication it forms a cyclic group of order 4 generated by $i$. It is closed under multiplication, contains the identity $1$, contains inverses ($i^{-1}=-i$, etc.), and is a subset of $(\mathbb{C}\setminus\{0\},\cdot)$. Therefore it is a subgroup of $(\mathbb{C}\setminus\{0\},\cdot)$.