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I'm reading Algebra 2e by Michael Artin. I'd be grateful for any guidance on 2 closely-related questions from Section 2.2, Groups and Subgroups, p. 42, excerpted below. First, I can understand the purpose and effect of the three rules listed, but I don't understand where they came from. Second, and similarly, I can see how the six elements of $S_{3}$ are non-overlapping and distinct, but I don't understand how they were derived. Where does the cancellation law fit in? (I'm going to press my luck and also ask how $x^{-1}$ in the first equation at the bottom is equal to $x^2$ in the second, by applying the first two rules). I am hoping these are simple and closely-enough related to hang together. If not, I would be glad to remove this and repost over a longer period. Cheers! Text:

To describe [$S_{3}$], we pick two particular permutations in terms of which we can write all others. We take the cyclic permutations ($\mathbf{1\ 2\ 3}$), and the transposition ($\mathbf{1\ 2}$), and label them as $x$ and $y$, respectively. The rules $x^3 = 1,\ y^2 = 1,\ yx = x^2y$ are easy to verify. Using the cancellation law, one sees that the six elements $1,\ x,\ x^2,\ y,\ xy,\ x^2y$ are distinct. So they are the six elements of the group: $S_{3} = \{1,\ x,\ x^2,\ y,\ xy,\ x^2y\}$. [...]

The rules [above] suffice for computation. Any product of the elements $x$ and $y$ and of their inverses can be shown to be equal to one of the products by applying the rules repeatedly. To do so, we move all occurrences of $y$ to the right side using the last rule, and we use the first two rules to keep the exponents small. For instance,

$x^{-1}y^3x^2y = x^2yx^2y = x^2(yx)xy = x^2(x^2y)xy = xyxy = x(x^2y)y = 1$.

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The reason $x^{-1}=x^2$ is that $1=x^3 = x^2x$; since the only element that when multiplied by $x$ yields $1$ is $x^{-1}$, it follows that $x{-1}=x^2$. The cancellation law comes in when showing the elements are distinct: if $x=x^2$, then cancelling we would get $x=1$, which is false; if $x^2=xy$, then cancelling $x$ we would get $x=y$, which is false. Etc. – Arturo Magidin Feb 15 '12 at 5:18
Got it - thanks. – Peter Feb 15 '12 at 5:39
up vote 3 down vote accepted

Do you understand how to multiply permutations? The rule $yx=x^2y$, for example, is just saying $$(1\ 2)(1\ 2\ 3)=(1\ 2\ 3)(1\ 2\ 3)(1\ 2)$$ and that is easy to verify if you know how to multiply permutations.

The cancellation law tells you, for example, that $x^2$ and $xy$ are distinct: if you had $x^2=xy$, then, by cancellation, you'd have $x=y$, which is manifestly false.

$x^3=1$ implies $(x)(x^2)=1$ which says $x^2$ is the inverse of $x$.

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