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_{3}

α = (1 2)

β = (2 3)

So I know that H={α

^{k}β

^{j}|j,k[itex]\in[/itex](the integers)}

However, would αβα or βαβ (in this case, they're equal) be in H?

The set H={ε,(1 2), (2 3), (1 2 3), (1 3 2)} (or {ε,α,β,αβ,βα})

would not be closed because (1 2 3)(1 2) = (1 3) which is not in H

But if (1 3) is in H you have all of S

_{3}which I thought was only generated by a 2-cycle and a 3-cycle.