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Math Help - Cyclic groups. Prove that <x^m> intersection <x^n> = <x^ (LCM,m,n)>

  1. #1
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    Cyclic groups. Prove that <x^m> intersection <x^n> = <x^ (LCM,m,n)>

    prove the following property of cyclic
    groups: Let G = <x> be a fi nite cyclic group.
    Prove that <x^m> intersection <x^n> = <x ^ LCM(m,n)>


    difficulty is that I have to account for situations where for example
    When |G| = 12, we have for example x^44 is in <x^20> intersection <x^14>= <x^140>

    attempt:

    Prove that <x^m> intersection <x^n> = <x^ (LCM,m,n)>

    let b be be in <x^m> intersection <x^n>
    then b = x^mk=x^nt for some k, t in Z
    then the exponent of b is a multiple of both m and n thus it's a multiple of Lcm(m,n), therefore b is in <x^LCM(m,n)

    now let b be in <x^LCM(m,n)>

    then b = x^LCM(m,n) *k for some k in Z
    Now LCM(m,n) = mt=nr
    then b = x^mtk=x^nrk thus b =x^m(tk)=x^n(rk) for some t,r in Z
    exponent of b is a multiple of both m & n so b is in the intersection of <x^m> and <x^n>

    I was told this is a quite difficult question
    and this seems too easy , which makes me think I'm missing something..
    thanks, Adam
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Cyclic groups. Prove that <x^m> intersection <x^n> = <x^ (LCM,m,n)>

    you are correct I cant see any problem in your solution
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