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Math Help - Order, periods, and cosets

  1. #1
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    Order, periods, and cosets

    Let G be a group of order 36 and let H be a normal subgroup of order 6. Let y \in G be an element of period 4.
    a) Explain why y\not\in H
    Because the period of any element must divide the order of the gorup that the element is in. 4 | 36 so y \in G but 4 does not divide 6 so y \not\in H

    b) What is the order of the factor group G/H
    G/H order is 6 because order H divides order G = 6

    c) Work out that (yH)^4 is trivial in G/H
    (yH)^4 = yHyHyHyH = yyyyHHHH = y^4H = eH, which is the trivial coset

    d) Prove yH has period 2 in G/H
    ...

    e) Prove that y^2 \in H
    ...

    For some reason when it comes to doing periods of cosets I get confused. Can anyone help me out?

    Thanks in advance,
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  2. #2
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    Quote Originally Posted by Zalren View Post
    Let G be a group of order 36 and let H be a normal subgroup of order 6. Let y \in G be an element of period 4.
    a) Explain why y\not\in H
    Because the period of any element must divide the order of the gorup that the element is in. 4 | 36 so y \in G but 4 does not divide 6 so y \not\in H

    b) What is the order of the factor group G/H
    G/H order is 6 because order H divides order G = 6

    c) Work out that (yH)^4 is trivial in G/H
    (yH)^4 = yHyHyHyH = yyyyHHHH = y^4H = eH, which is the trivial coset

    d) Prove yH has period 2 in G/H


    Neither y\in H (why?) nor ord(yH)=1 (why?These two claims are, in

    fact, one and the same thing...), but still (yH)^2=1\in G/H, thus...




    e) Prove that y^2 \in H


    This follows at once from (c) and from xH=H\Longleftrightarrow x\in H

    Tonio



    For some reason when it comes to doing periods of cosets I get confused. Can anyone help me out?

    Thanks in advance,
    .
    Follow Math Help Forum on Facebook and Google+

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