Proving Normal Subgroup of S4 in Alternative Ways

[msd213]

How would one go about proving a particular subset of S4 is a normal subgroup of S4? Since S4 has 24 elements, I'm wondering if there is any other way to prove this other than a brute force method.

 

[tauon]

you can use a certain (obvious) implication of the Lagrange theorem for groups to thin out the workload; and you can also use the fact that any permutation is a product of disjoint transpositions to further simplify notation... of course if by S4 you mean S_4 the symmetry group.

so yes there is a way to find the normal subgroups by other means than brute force.

 

[msd213]

you can use a certain (obvious) implication of the Lagrange theorem for groups to thin out the workload; and you can also use the fact that any permutation is a product of disjoint transpositions to further simplify notation... of course if by S4 you mean S_4 the symmetry group.

so yes there is a way to find the normal subgroups by other means than brute force.

Yes, the symmetry group.

I'm not sure I entirely know what you mean with Lagrange's theorem. I'm not sure how that exactly helps.

And breaking up each permutation into transpositions, won't that just make the brute force easier?

 

[msd213]

you can use a certain (obvious) implication of the Lagrange theorem for groups to thin out the workload; and you can also use the fact that any permutation is a product of disjoint transpositions to further simplify notation... of course if by S4 you mean S_4 the symmetry group.

so yes there is a way to find the normal subgroups by other means than brute force.

Yes, the symmetry group.

I'm not sure I entirely know what you mean with Lagrange's theorem. I'm not sure how that exactly helps.

And breaking up each permutation into transpositions, won't that just make the brute force easier?