The definition of semidirect group on Rotman's Introduction to the theory of groups is the following:
A group $G$ is a semidirect product of $K$ by $Q$, denoted by $G=K \rtimes Q$, if $K \lhd G$ and $K$ has a complement $Q_1 \cong Q$.
The definition of complement is this one:
If $K \leq G$, $Q \leq G$ is a complement of $K$ if their intersection is trivial and $KQ = G$.
There are a few things I do not quite understand here.
I know that, if $K$ has complements, they're all isomorphic to each other; now, if a subgroup of $G$ is isomorphic to a complement of $K$, will it also be a complement for it? In other words, if $KQ = G$ and $Q_1 \cong Q$, is it also $KQ_1 = G$?
$Q_1$ in the definition of semidirect product is a subgroup of $G$, because it's a complement for $K$ and all complements are, by definition, subgroups of $G$. However, all we know about $Q$ is that it's isomorphic to $Q_1$; it may well be not even a subset of $G$, which means $G = K \rtimes Q$ can't be a strict set-theoretical equivalence. Why bother with $Q$? Why not just $G = K \rtimes Q_1$?
Thanks in advance!