Is the Generated Set of a Group Closed Under Group Operation?

[BrockDoiron]

Hi, I am told to give the subgroup H=<α,β> with α,β$\in$S₃

α = (1 2)
β = (2 3)

So I know that H={α^kβ^j|j,k$\in$(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.

 

[HallsofIvy]

Yes, a group is closed under the group operation so any combinations of $\alpha$ and $\beta$ must also be in the group.