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Math Help - Sylow Subgroups

  1. #1
    Member Jason Bourne's Avatar
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    Sylow Subgroups

    I'm not sure with the following:

    (1) What is the order of every Sylow 5-subgroup of A_5 \times A_5 ?

    (2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of G \times H has the form P_1 \times Q_1 where P_1 is a sylow p-subgroup of G, and Q_1 is a sylow p-subgroup of H.

    (3) Find the number of sylow 5-subgroups of A_5 \times A_5
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    For (1) i think it's 25, as its's 5.5 ?

    For (2) I think I use the Conjugacy sylow theorem somehow?

    For (3) I think it might be 26 because A_5 \times A_5 is simple?

    Thanks!
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by Jason Bourne View Post
    I'm not sure with the following:

    (1) What is the order of every Sylow 5-subgroup of A_5 \times A_5 ?

    (2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of G \times H has the form P_1 \times Q_1 where P_1 is a sylow p-subgroup of G, and Q_1 is a sylow p-subgroup of H.

    (3) Find the number of sylow 5-subgroups of A_5 \times A_5
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    For (1) i think it's 25, as its's 5.5 ?

    For (2) I think I use the Conjugacy sylow theorem somehow?

    For (3) I think it might be 26 because A_5 \times A_5 is simple?

    Thanks!
    Hi Jason Bourne.

    (1) Your answer is correct.

    (2) Determine the maximum power of p dividing the order of G\times H.

    (3) No, it can’t be 26. The order of A_5\times A_5 is 3600 = 2^4\times3^2\times5^2. The number of Sylow 5-subgroups must divide 2^4\times3^2=144. 26 does not divide 144.
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  3. #3
    MHF Contributor

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    for part (3) use part (2) and the fact that A_5 has 6 Sylow 5-subgroups. to prove that P \times Q is a Sylow p-subgroup of G \times H if P is a Sylow p-subgroup of G and Q is a Sylow p-subgroup H

    do as TheAbstractionist suggested. but the converse is less trivial and it's the only fairly interesting part of your problem!
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