Proving H = S_4: Lagrange's Theorem

[math_nerd]

Suppose that H is a subset of S_4, and that H contains (12) and (234). Prove that H=S_4.

The order of S_4 is 24.

(12)(234) = (1342) shows that the finite subset of group H is closed under the group operation, so it is a subgroup of H.

Then, using Lagrange's Theorem, |S_4|=24/2 = 12. so the order of H is 12 by this.

And now I am lost on how I can prove that H = S_4.

 

[micromass]

Suppose that H is a subset of S_4, and that H contains (12) and (234). Prove that H=S_4.

There is something wrong here. The problem, as stated, is false. Did you mean that H has to be a subgroup of S_4? Because this is probably true...

 

[math_nerd]

* H is a subgroup.

And how is this true? I'm pretty lost on how to show this.