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I understand the standard definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows:

Let $H\leq G$. Then a left coset of $H$ is a nonempty set $S\subset G$ such that $x^{-1}y \in H$ for any $x,y\in S$, and for any fixed $x\in S$, the map $y\mapsto x^{-1}y $ is a surjection from $S$ to $H$.

I see how it works, but it doesn't seem to say much... What am I not seeing?

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up vote 6 down vote accepted

I'm not exactly sure what you mean by the canonical definition, but one standard definition of a coset is a translate $xH$ of $H$, for some $x\in G$. Another definition is that a coset is an equivalence class of the equivalence relation $\sim$ given by $x\sim y$ if $x^{-1}y\in H$. This is, in spirit, close to the definition you quote.

The equivalence class definition is useful in a couple of ways.

First, it's good to know that $\sim$ really is an equivalence relation, because that tells you immediately that the cosets partition $G$.

Second, it gives you a way of checking when two elements $x$ and $y$ represent the same coset: simply check if $x^{-1}y\in H$. This is similar to the idea of checking when two numbers $a$ and $b$ have the same remainder mod $m$: check if $m$ divides $a-b$.

The definition you quoted connects the two standard definitions -- the definition in terms of translates and the definition in terms of equivalence classes. Indeed, the fact that the transformation $y\mapsto x^{-1}y$ from $S$ to $H$ is surjective tells us that if we translate $S$ by $x^{-1}$ we get $H$ back.

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