let $A$ and $B$ be two subgroups of $G$. we say that $B$ is a complement of $A$ if :
$G=AB$
$A\cap B=\{1\}$
Given a subgroup $A$ of $G$ i don't see how the complement $B$ of $A$ in $G$ is not unique, it seems to me like $A$ and $B$ partition $G$ right? I mean with these two conditions an element in $G$ must be lying in $A$ or in $B$, that is the subgoup $A$ has a complement if $(G-A)\cup\{1\}$ is also a subgroup of $G$ ?