We have already seen one way we can examine a complicated group $G\text{:}$ namely, study its subgroups, whose group structures are in some cases much more directly understandable than the structure of $G$ itself. But if $H$ is a subgroup of a group $G\text{,}$ if we only study $H$ we lose all the information about $G$'s structure “outside” of $H\text{.}$ We might hope that $G-H$ (that is, the set of elements of $G$ that are not in $H$) is also a subgroup of $G\text{,}$ but we immediately see that cannot be the case since the identity element of $G$ must be in $H\text{,}$ and $H\cap (G-H)=\emptyset\text{.}$ Instead, let's ask how we can get at some understanding of $G$'s entire structure using a subgroup $H\text{?}$ It turns out we use what are called cosets of $H\text{;}$ but before we get to those, we need to cover some preliminary material.