Proving Normal Subgroup of S4 in Alternative Ways

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SUMMARY

To prove that a specific subset of S4 is a normal subgroup, one can utilize Lagrange's theorem and the property that every permutation can be expressed as a product of disjoint transpositions. These methods significantly reduce the complexity of the proof compared to brute force techniques. The discussion emphasizes that understanding the implications of Lagrange's theorem is crucial for simplifying the workload in subgroup verification within the symmetry group S4.

PREREQUISITES
  • Understanding of Lagrange's theorem in group theory
  • Familiarity with permutation groups, specifically S4
  • Knowledge of transpositions and their role in permutations
  • Basic concepts of normal subgroups
NEXT STEPS
  • Study the implications of Lagrange's theorem for group theory
  • Learn about the structure and properties of the symmetric group S4
  • Explore the concept of normal subgroups in greater depth
  • Investigate methods for simplifying proofs in group theory
USEFUL FOR

Mathematicians, particularly those specializing in group theory, educators teaching abstract algebra, and students preparing for advanced studies in algebraic structures.

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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.
 
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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 S_4 the symmetry group.

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