Show S4 has only one unique subgroup of order 12

[linzedun]

Question: Show that S4 has a unique subgroup of order 12.

 

So far I know that A4 is a subgroup of order 12. So I was trying to prove by contradiction that there was another subgroup of order 12, say G. So the (|G||A4|/the intersection of G and A4) = 24. Then I need to show something to the effect that the intersection of G and A4 has to be a subgroup of order 6 in A4 and that that is a contradiction.

 

This is as far as I got and my teacher says I am on the right track, but I have no ideas of where to go next, help please!

[matt grime]

Hint G is also normal.

[linzedun]

So I can prove that A4 has no subgroup of order 6 and then I have it proved right?

[matt grime]

No, not as stands. Proving A4 has no subgroup of order six isn't going to help unless you have provem that the assumption G exists and isn't A4 implies that it must possess such. You don't appear to have done that.