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Math Help - Finding a subgroup

  1. #1
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    Finding a subgroup

    Hi,

    started a topic on subgroups this week and I can't seem to find any examples to help me with a couple of questions.

    The first one is 'Find all the subgroups of Z30' I'm not to sure if this is just the factors of 30 am I right?

    The second part of the question I'm stuck on is not necessarily about subgroups (or maybe it is, not to sure!) but it asks to determine the order of every element in (i) D8 and (ii). D12 .

    Would appreciate very much if anyone could help me, thanks!
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Finding a subgroup

    some things you should be aware of (and might try to prove, if you feel ambitious):

    1) every subgroup of a cyclic group is cyclic.

    2) if G is cyclic of order n, and k is a divisor of n, G has EXACTLY one subgroup of order k.

    in particular, (2) indicates your hunch is correct. the subgroups of Z30 are:

    <0> = {0}
    <1> = Z30 (and also equals <7>, <11>, <13>, <17>, <19>, <23>, and <29>)
    <2> = {0,2,4,6,8,10,12,14,16,18,20,22,24,26,28} (this is also equal to <4>, <8>, <14>, <16>, <22>, 26>, and <28>)
    <3> = {0,3,6,9,12,15,18,21,24,27} (this is also equal to <9>, <21>, and <27>)
    <5> = {0,5,10,15,20,25} (this is also equal to <25>)
    <6> = {0,6,12,18,24} (this is also equal to <12>, <18>, and <24>)
    <10> = {0,10,20} (this also equals <20>)
    <15> = {0,15}

    we can prove some results about D2n, the dihedral group of order 2n. note that this can be presented as <r,s> where:

    rn = e
    s2 = e
    sr = r-1s.

    you can prove by induction on |k| that:

    srk = r-ks for all integers k.

    this means all elements of the form rks have order 2:

    (rks)2 = (rks)(rks) = rk(srk)s = rk(r-ks)s = s2 = e.

    all the other elements are in <r>, and can be analyzed like elements of any cyclic group.
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