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Thread: Problem re: Sylow's Theorems

  1. #1
    r45
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    Problem re: Sylow's Theorems

    Hi there,

    I had this question on an exam today and I'm not sure how to do it.

    "Let G = $\displaystyle \mathbb{Z}_3$ x $\displaystyle A_4$. List all the Sylow 2-subgroups of G, and all the Sylow 3-subgroups of G, giving reasons that your lists are complete"

    I found that G has order 36, and so I determined that $\displaystyle n_2$ = 1, 3 or 9 and that $\displaystyle n_3$ = 1 or 4 (where $\displaystyle n_i$ denotes the number of Sylow i-subgroups) but from there I couldn't do much. By brute force I managed to find one subgroup of order 4 and one of order 9, so in the end I just put that $\displaystyle n_2$ = 1 and $\displaystyle n_3$= 1.

    Can anyone verify, or correct this?

    Many thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by r45 View Post
    Hi there,

    I had this question on an exam today and I'm not sure how to do it.

    "Let G = $\displaystyle \mathbb{Z}_3$ x $\displaystyle A_4$. List all the Sylow 2-subgroups of G, and all the Sylow 3-subgroups of G, giving reasons that your lists are complete"

    I found that G has order 36, and so I determined that $\displaystyle n_2$ = 1, 3 or 9 and that $\displaystyle n_3$ = 1 or 4 (where $\displaystyle n_i$ denotes the number of Sylow i-subgroups) but from there I couldn't do much. By brute force I managed to find one subgroup of order 4 and one of order 9, so in the end I just put that $\displaystyle n_2$ = 1 and $\displaystyle n_3$= 1.

    Can anyone verify, or correct this?

    Many thanks
    $\displaystyle A_4$ has four subgroups of order $\displaystyle 3$. For example, $\displaystyle \left|\langle (1,2,3)\rangle\right|=3$. It follows then that $\displaystyle n_3=4$ (these can actually be all found by using the fact that all Sylow $\displaystyle p$-subgroups are conjugate). Similarly, there is one subgroup of order $\displaystyle 4$ of $\displaystyle A_4$ namely $\displaystyle \{e,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$ from where it pretty easily follows that $\displaystyle n_2=1$.
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  3. #3
    r45
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    I thought that when looking for $\displaystyle n_3$ I was trying to find subgroups of order 9? Because $\displaystyle 36 = 3^{2}.4$, so in this case Sylow 3-subgroups have order $\displaystyle 3^{2}=9$?
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  4. #4
    r45
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    So to clarify on our Sylow 2-subgroup in the context of G as I defined above, would it be:

    $\displaystyle \{(1, (e)), (1, (13)(24)), (1, (12)(34)), (1, (14)(23))\}$?

    By the way thankyou very much for the response.
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    Quote Originally Posted by r45 View Post
    I thought that when looking for $\displaystyle n_3$ I was trying to find subgroups of order 9? Because $\displaystyle 36 = 3^{2}.4$, so in this case Sylow 3-subgroups have order $\displaystyle 3^{2}=9$?

    You're right. Drexel apparently misread and thought you were talking of $\displaystyle A_4$ and not of $\displaystyle G:=\mathbb{Z}_3\times A_4$...or perhaps

    he thought to mention the 3-Sylow sbgps. of $\displaystyle A_4$ as a hint for you.

    Anyway, his description gives you some hints: any 3-Sylow sbgp. of G will be the product of $\displaystyle \mathbb{Z}_3$ with a sbgp. of order 3 of $\displaystyle A_4$ ...

    Tonio
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tonio View Post
    or perhaps

    he thought to mention the 3-Sylow sbgps. of $\displaystyle A_4$ as a hint for you.

    Anyway, his description gives you some hints: any 3-Sylow sbgp. of G will be the product of $\displaystyle \mathbb{Z}_3$ with a sbgp. of order 3 of $\displaystyle A_4$ ...

    Tonio
    This is what I was doing, sorry if there was any confusion!
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  7. #7
    MHF Contributor Swlabr's Avatar
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    You can prove there is one Sylow 2-subgroup by proving that your subgroup of order 4 is normal; in general, once you have a maximal subgroup of order $\displaystyle p^i$, conjugate it with everything you can find to find the other Sylow p-subgroups...
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