Generating Set of a Group

  • #1
Hi, I am told to give the subgroup H=<α,β> with α,β[itex]\in[/itex]S3

α = (1 2)
β = (2 3)

So I know that H={αkβj|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 S3 which I thought was only generated by a 2-cycle and a 3-cycle.
 

Answers and Replies

  • #2
HallsofIvy
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Yes, a group is closed under the group operation so any combinations of [itex]\alpha[/itex] and [itex]\beta[/itex] must also be in the group.
 

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