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Math Help - find an integer k such that the subgroup is <k>

  1. #1
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    Question find an integer k such that the subgroup is <k>

    The smallest subgroup containing a collection of elements S is the
    subgroup H with the property that if K is any subgroup containing
    S then K also contains H. (So, the smallest subgroup containing S is
    contained in every subgroup that contains S.) The notation for this
    subgroup is <S>. In the group Z, find
    a. <8, 14>
    b. <8, 13>
    c. <6, 15>
    d. <m, n>
    e. <12, 18, 45>.
    In each part, find an integer k such that the subgroup is k.
    Attached Thumbnails Attached Thumbnails find an integer k such that the subgroup is &lt;k&gt;-untitled.png  
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  2. #2
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    Re: find an integer k such that the subgroup is <k>

    Quote Originally Posted by MathGeni04 View Post
    The smallest subgroup containing a collection of elements S is the
    subgroup H with the property that if K is any subgroup containing
    S then K also contains H. (So, the smallest subgroup containing S is
    contained in every subgroup that contains S.) The notation for this
    subgroup is <S>. In the group Z, find
    a. <8, 14>
    So the smallest set of integers containing 8 and 14? That would, of course, include 0 as well as 8+ 8= 16 and, in fact, any multiple of 8 or 14. It would include 8+ 14= 22, 14- 8= 6, etc. In fact it would include any x= 8n+ 14m= 2(4n+ 7m) for any integers n and m. (So, first, it must include only even integers.) What can you say about numbers of the form 4n+ 7m? One of the results of the "Euclidean Algorithm" is that since 4 and 7 are "relatively prime", there exist m and n such that 4n+ 7m= x for all integers x.

    b. <8, 13>
    8 and 13 are relatively prime.

    c. <6, 15>
    6= 3(2), 15= 3(5) so 6m+ 15n= 3(2m+ 5n).

    d. <m, n>
    Is no information at all given about m and n?

    [quote]e. <12, 18, 45>.[quote]
    12= 2(2)(3), 18= 2(3)(3), 45= 3(3)(5). The greatest common divisor is 3.

    In each part, find an integer k such that the subgroup is k.
    You mean "the subgroup is <k>."
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  3. #3
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    Re: find an integer k such that the subgroup is <k>

    Hi,
    The attachment expands a little on the previous response. I hope if you read and understand this, you can answer all your questions easily. For the last question e, you just have to note that <a,b,c>=<<a,b>,c>.

    find an integer k such that the subgroup is &lt;k&gt;-mhfdiscretemath6a.png
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