Proving H = S_4: Lagrange's Theorem

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SUMMARY

The discussion centers on proving that the subgroup H of the symmetric group S_4, which contains the elements (12) and (234), is equal to S_4. The order of S_4 is established as 24, and through the application of Lagrange's Theorem, it is determined that the index |S_4:H| equals 12, indicating that the order of H is 12. However, the conclusion that H equals S_4 is challenged, as the initial premise may be flawed, suggesting that H must indeed be a subgroup of S_4 rather than equal to it.

PREREQUISITES
  • Understanding of group theory concepts, specifically symmetric groups.
  • Familiarity with Lagrange's Theorem in group theory.
  • Knowledge of group operations and closure properties.
  • Basic understanding of subgroup definitions and properties.
NEXT STEPS
  • Study the properties of symmetric groups, particularly S_4.
  • Review Lagrange's Theorem and its implications for subgroup orders.
  • Explore examples of subgroup generation in symmetric groups.
  • Investigate the closure properties of groups and their significance in subgroup proofs.
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This discussion is beneficial for students and educators in abstract algebra, particularly those focusing on group theory, as well as mathematicians interested in the properties of symmetric groups and subgroup analysis.

math_nerd
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Suppose that H is a subset of S_4, and that H contains (12) and (234). Prove that H=S_4.

The order of S_4 is 24.

(12)(234) = (1342) shows that the finite subset of group H is closed under the group operation, so it is a subgroup of H.

Then, using Lagrange's Theorem, |S_4:H|=24/2 = 12. so the order of H is 12 by this.

And now I am lost on how I can prove that H = S_4.
 
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math_nerd said:
Suppose that H is a subset of S_4, and that H contains (12) and (234). Prove that H=S_4.

There is something wrong here. The problem, as stated, is false. Did you mean that H has to be a subgroup of S_4? Because this is probably true...
 
* H is a subgroup.

And how is this true? I'm pretty lost on how to show this.
 

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