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Math Help - Non-abelian Group of order 8 problem

  1. #1
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    Non-abelian Group of order 8 problem

    Heres the problem:

    let G be a non-abelian group of order 8

    1. prove there exists an element of G, call it a such that it's order is 4.
    2. Let b be an element of G such that b is not formed by a. Show that a^{-1}=a^3=b^{-1}ab.
    3. Show that if the order of b is 2 then G is isomorphic to D_{4}, and if the order of b is 4 then G is isomorphic to the quatrernion group.

    My solution so far:

    I was able to show 1 easily by using legrange's theorem and eliminating the possiblities of 8,1,2 being the only order of elements.

    Since a is of order 4 it is obvious why a^{-1}=a^3.

    To show 2 I showed that the elements of G must be e,a,a^2,a^3,b,ab,a^2b,a^3b. Then my idea is to show by elimination that ba^3=ab.

    I was able to eliminate all options besides ba^3=a^2b and this is where I'm stuck.

    I'm not sure if this is the right approach at all or is there a much simpler and elegant way of showing this.

    Thanks,

    SK
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  2. #2
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    Quote Originally Posted by skyking View Post
    Heres the problem:

    let G be a non-abelian group of order 8

    1. prove there exists an element of G, call it a such that it's order is 4.
    2. Let b be an element of G such that b is not formed by a. Show that a^{-1}=a^3=b^{-1}ab.
    3. Show that if the order of b is 2 then G is isomorphic to D_{4}, and if the order of b is 4 then G is isomorphic to the quatrernion group.

    My solution so far:

    I was able to show 1 easily by using legrange's theorem and eliminating the possiblities of 8,1,2 being the only order of elements.

    Since a is of order 4 it is obvious why a^{-1}=a^3.

    To show 2 I showed that the elements of G must be e,a,a^2,a^3,b,ab,a^2b,a^3b. Then my idea is to show by elimination that ba^3=ab.

    I was able to eliminate all options besides ba^3=a^2b and this is where I'm stuck.


    ba^3=a^2b\Longrightarrow a^3=b^{-1}a^2b\Longrightarrow a^2=(a^3)^2=b^{-1}a^2bb^{-1}a^2b=b^{-1}a^4b=1\Longrightarrow contradiction.

    Tonio



    I'm not sure if this is the right approach at all or is there a much simpler and elegant way of showing this.

    Thanks,

    SK
    .
    Follow Math Help Forum on Facebook and Google+

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