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Math Help - group of order pq

  1. #1
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    Lightbulb group of order pq

    I have a question concerning a group of order pq, where p,q are distinct prime numbers.

    Does such a group necessarily have a normal subgroup of order p, and a normal subgroup of order q?


    Thank you !
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  2. #2
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    Yes, this follows from Sylow's Third Theorem. If n is the number of Sylow p-subgroups in G, then n = 1 (mod p) and n|pq. However, the only value for n that satisfies both of these conditions is 1. So, there is only one Sylow p-subgroup in G, meaning it is normal. The same is true for the single Sylow q-subgroup.
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  3. #3
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    Quote Originally Posted by spoon737 View Post
    Yes, this follows from Sylow's Third Theorem. If n is the number of Sylow p-subgroups in G, then n = 1 (mod p) and n|pq. However, the only value for n that satisfies both of these conditions is 1. So, there is only one Sylow p-subgroup in G, meaning it is normal. The same is true for the single Sylow q-subgroup.

    This isn't quite accurate: there will always be a normal Sylow subgroup of order the greatest prime, but the Sylow sbgp. of order the lowest prime may not be normal. For example, S_3 , of order 6=2\cdot 3 , has a normal subgroup of order 3 but no normal sbgp. of order 2.

    Tonio
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  4. #4
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    Quote Originally Posted by tonio View Post
    This isn't quite accurate: there will always be a normal Sylow subgroup of order the greatest prime, but the Sylow sbgp. of order the lowest prime may not be normal. For example, S_3 , of order 6=2\cdot 3 , has a normal subgroup of order 3 but no normal sbgp. of order 2.

    Tonio
    Good point, I was a little too hasty in my answer.
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  5. #5
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    Quote Originally Posted by spoon737 View Post
    Yes, this follows from Sylow's Third Theorem. If n is the number of Sylow p-subgroups in G, then n = 1 (mod p) and n|pq. However, the only value for n that satisfies both of these conditions is 1. So, there is only one Sylow p-subgroup in G, meaning it is normal. The same is true for the single Sylow q-subgroup.
    Not correct. Even if p,q are both odd, e.g. p=5,q=11, you could have a group with 11 Sylow 5-subgroups. However it's true that there's only one Sylow q-subgroup if q>p
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  6. #6
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    thank you.
    It will take some time for me to understand what you said, but thank you
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