Proving that H=S_4 for H subset of S_4

[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.
The order of H is 2.

(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.

 

[lanedance]

The order of H is 2.

why is it 2? how about looking at compositions of the multiplications and inverses that must be contained in H for it to be a group

 

[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.

The previous post had a mistake. Sorry about that.

Now, to show that H is a subgroup of K: look at compositions of multiplications and inverses in H for it to be a group? How do I do this? I'm sorry this probably sounds really dumb, but all I know is that, aH=Ha iff H=aHa^-1 (property of cosets). I just don't see how this is done, and whatvthat will show.

 

[lanedance]

does is say H is a subgroup?