Normal Subgroup of S4

  • Thread starter msd213
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  • #1
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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.
 

Answers and Replies

  • #2
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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 [tex]S_4[/tex] the symmetry group.

so yes there is a way to find the normal subgroups by other means than brute force.:smile:
 
  • #3
25
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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 [tex]S_4[/tex] the symmetry group.

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

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?
 
  • #4
25
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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 [tex]S_4[/tex] the symmetry group.

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

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?
 

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