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Recently I was going over online notes regarding symmetry groups and I came across the following notation:

$S_3=\{1,x,x^2,y,xy,x^2y\}$ is generated by $\{x,y\}$. What does this mean? Aren't the elements in $S_3$ of the form $\{(12),(123),(23),(132),e, (13)\}$. Can someone please explain?

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  • $\begingroup$ the first notation is usually used when speaking about dihedral group (when most of times $\sigma=x,\tau=y$). The diheral group has geometric meaning of reflection and rotation. If you take triangle, number its vertexes, and permute them according to $S_3$ it would be the same to activate on it, the appropriating rotation and reflection from $D_3$. $\endgroup$ – user65985 Oct 17 '13 at 2:35
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Take $x=(123)$ and $y=(12)$. Note that $x^3=e$ and $y^2=e$.

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  • $\begingroup$ But why do that substitution? Does it have a geometric meaning at all? $\endgroup$ – Rutherford Mark Oct 17 '13 at 2:29
  • $\begingroup$ @RutherfordMark, yes, $S_3=D_6$, the symmetric group of the equilateral triangle. With this interpretation, $x$ is rotation by $120^\circ$ and $y$ is reflection along a median line. $\endgroup$ – lhf Oct 17 '13 at 2:31

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