if $Q=[{+-1,+-i,+-j,+-k}]$ , i.e. the quaternions, then what would be two examples of a minimal generating set for Q?.One would be {i} right ? Because :
$i^2=j^2=k^2=-1$ , $(-1)^2=e=1 \Rightarrow (-1)^{-1}=1 $
and
$i^2=-1 \Rightarrow i^4=1 \Rightarrow (i)^{-1}=i^3$
$j^2=-1 \Rightarrow j^4=1 \Rightarrow (j)^{-1}=j^3$
$k^2=-1 \Rightarrow k^4=1 \Rightarrow k^{-1}=k^3$
so we have the relations $i^2=-1=j^2=k^2$ and $i^4=1=j^4=k^4 \Rightarrow i^{-1}=-1=i^3=j^3=k^3$ ans so our set has been generated using the generator i and its inverse , correct ? So another such set would b {j} or {k) etc...
The quaternion group is not cyclic, so it needs at least $2$ generators. On the other hand, two generators are sufficient, see here: $$Q_8=\langle a,b \mid a^4=1,b^2=a^2, b^{-1}ab=a^{-1}\rangle .$$