Abstract Algebra | Converse of Lagrange's Theorem

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We prove that the converse of Lagrange's Theorem is not true by proving that the alternating group A_4 has no subgroup of order 6.

  1. Michael Penn
  2. math
  3. mathematics
  4. number theory
  5. abstract algebra
  6. calculus
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Question: If gH = Hg only for g not in H how does that mean all conjugates of elements of H are in H? Wouldn't it just show that conjugates of H are H for g not in H? So in your proof we would have to establish that the elements you're using to transform each cycle are not in H. True? Or am I missing something?

FrankBria