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Tamal Krishnan posted an Question
October 14, 2021 • 16:26 pm 30 points
  • IIT JAM
  • Mathematics (MA)

Prove that a non commutative group of order 10 must have a subgroup order 5

prove that a non commutative group of order 10 must have a subgroup order 5

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    Deepak singh 1

    Let G be a group of order 10. By Lagrange's theorem, if there exist a subgroup H of G, then o(H)=1,2,5 or 10. Assume that there is no subgroup of order 5. Let, e≠a∈G. Then we have o(a)|o(G). If every element of G is of order 2, then a^2=e i.e a=a^−1 for all a∈G. Implies that G is abelian. This is a contradiction. Therefore, every element of G is not of order 2. Thus, o(a)=5 or 10. If o(a)=5, then H=⟨a⟩ is a cyclic subgroup of order 5.

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