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Math Help - Modern Algebra Problem

  1. #1
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    Modern Algebra Problem

    Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.

    Can I say that pq = n є Z < ∞?
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  2. #2
    Senior Member Tinyboss's Avatar
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    The order of a subgroup divides the order of the group. Groups of prime order are cyclic.
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  3. #3
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    For a slightly more descriptive answer, Lagrange's theorem says that for finite groups, a subgroup's order divides that of the group. Since the group's order is a product of two primes, a nontrivial subgroup should have order p, q or 1. Groups of prime order are cyclic, because in particular, the subgroup generated by any non-identity element must be the whole thing, as it would have order dividing a prime number.
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  4. #4
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    Well, I might be wrong, but the condition should be "proper" subgroup. There are groups of order pq that are non-abelian.
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  5. #5
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    Thank you everyone

    I've solved the problem.
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