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Thread: 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 $\displaystyle A_5 \times A_5$ ?

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

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

    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 $\displaystyle A_5 \times A_5$ 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 $\displaystyle A_5 \times A_5$ ?

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

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

    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 $\displaystyle A_5 \times A_5$ is simple?

    Thanks!
    Hi Jason Bourne.

    (1) Your answer is correct.

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

    (3) No, it can’t be 26. The order of $\displaystyle A_5\times A_5$ is $\displaystyle 3600 = 2^4\times3^2\times5^2.$ The number of Sylow 5-subgroups must divide $\displaystyle 2^4\times3^2=144.$ 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 $\displaystyle A_5$ has 6 Sylow 5-subgroups. to prove that $\displaystyle P \times Q$ is a Sylow p-subgroup of $\displaystyle G \times H$ if $\displaystyle P$ is a Sylow p-subgroup of $\displaystyle G$ and $\displaystyle Q$ is a Sylow p-subgroup $\displaystyle H$

    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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