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Chapter7The Wonderful World of Cosets

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.