Show that S4 (the symmetric group of degree 4) has a unique subgroup of order 12.

I know that A4 is that subgroup but I'm not really sure how to show that it is the unique subgroup. Help?

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- December 3rd 2013, 06:36 PMkellsbells92S4 subgroup of order 12
Show that S4 (the symmetric group of degree 4) has a unique subgroup of order 12.

I know that A4 is that subgroup but I'm not really sure how to show that it is the unique subgroup. Help? - December 4th 2013, 09:55 AMjohngRe: S4 subgroup of order 12
Hi,

I'm tired of trying to write latex in this editor. So the hints are in an attachment. If you have problems, post them.

Attachment 29855 - December 4th 2013, 11:20 AMkellsbells92Re: S4 subgroup of order 12
thank you!

- December 16th 2013, 02:17 PMDevenoRe: S4 subgroup of order 12
We can approach this another way:

Consider the homomorphism .

If is ANY subgroup of , then is also a subgroup of . Restricting the homomorphism sgn to this subgroup yields a homomorphism from to {-1,1}.

Since there are only two possibilities for the image of this homomorphism, we have either:

, which implies that is a subgroup of (why?), or:

, which implies that is of index 2 in .

Now if , the first possibility leads to . So if , we must have that .

Now is thus a subgroup of of order 6. There are two possibilities:

a) This subgroup is cyclic, but has no elements of order 6, which leaves us with:

b) .

However, if has a subgroup isomorphic to , this subgroup would contain 3 elements of order 2. However, the (only) 3 elements of order 2 in , namely:

(1 2)(3 4), (1 3)(2 4), (1 4)(2 3) generate a subgroup of order 4, and has no such subgroup (it cannot, for 4 does not divide 6).