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Math Help - cyclic group help

  1. #1
    Senior Member Danneedshelp's Avatar
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    cyclic group help

    Hello, I want to make sure I understand the fundamental theorem of cyclic groups correctly.

    For example, consider the group (\mathbb{Z}_{45},+_{45}). I need to construct a subgroup lattice for (\mathbb{Z}_{45},+_{45}). Since the order of (\mathbb{Z}_{45},+_{45}) is 45, I need to find all the natural numbers k that divide 45. Of course, these numbers are 1, 3, 5, 9, 15, and 45, which have orders 45, 15, 9, 5, 3, and 1, respectively. Since <1>=(\mathbb{Z}_{45},+_{45}) we have <1^{5}>=<5>, <1^{9}>=<9>, <1^{15}>=<15>, and so on are the subgroups of (\mathbb{Z}_{45},+_{45}). From here I just decide which set is a subset of the other and I am done.

    Am I using the theorem correctly? Does this mean that any natural number k that does not divide the order of (\mathbb{Z}_{45},+_{45}) will create set that does not abide by group axioms? I know I can check this for the cyclic group in this example, but, in general, is the previous sentence correct?
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    Hello, I want to make sure I understand the fundamental theorem of cyclic groups correctly.

    For example, consider the group (\mathbb{Z}_{45},+_{45}). I need to construct a subgroup lattice for (\mathbb{Z}_{45},+_{45}). Since the order of (\mathbb{Z}_{45},+_{45}) is 45, I need to find all the natural numbers k that divide 45. Of course, these numbers are 1, 3, 5, 9, 15, and 45, which have orders 45, 15, 9, 5, 3, and 1, respectively. Since <1>=(\mathbb{Z}_{45},+_{45}) we have <1^{5}>=<5>, <1^{9}>=<9>, <1^{15}>=<15>, and so on are the subgroups of (\mathbb{Z}_{45},+_{45}). From here I just decide which set is a subset of the other and I am done.

    Am I using the theorem correctly? Does this mean that any natural number k that does not divide the order of (\mathbb{Z}_{45},+_{45}) will create set that does not abide by group axioms? I know I can check this for the cyclic group in this example, but, in general, is the previous sentence correct?

    All is fine, but if you write \mathbb{Z}_{45} as an ADDITIVE group, then the subgroups are 5\mathbb{Z}_{45}=\{0,5,10,15,20,25,30,35,40\}= the subgroup of order 9,

    9\mathbb{Z}_{45}=\{0,9,18,27,36\}= the subgroup of order 5, etc.

    After all, here 1=1^5=1^{15}\!\!\pmod {45}

    Tonio
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