# Normal Subgroup of S4

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.

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. 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?

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?