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Math Help - Exact Sequences - Diagrams that 'commute' - Example

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    Super Member Bernhard's Avatar
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    Exact Sequences - Diagrams that 'commute' - Example

    I am reading Dummit and Foote Section 10.5 on Exact Sequences.

    I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example.

    The example, as you can no doubt see, requires an understanding of the nature of the quotient module  (\mathbb{Z} / m  \mathbb{Z} ) / (n \mathbb{Z} / m \mathbb{Z} )

    To make this quotient more tangible, in this example take m = 6, n = 3 so k = 2.

    Then we are trying to understand the nature of the quotient module  (\mathbb{Z} / 6  \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} )

    Now consider the nature of   (\mathbb{Z} / 6  \mathbb{Z} )

    We have  0 + \mathbb{Z} / 6  \mathbb{Z}   = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }

    and  1 + \mathbb{Z} / 6  \mathbb{Z}    = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }

    and so on

    But what is  3 \mathbb{Z} / 6 \mathbb{Z}  ? and indeed, further, what is  (\mathbb{Z} / 6  \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) ?

    Can someone please help clarify this matter?

    Peter
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    Re: Exact Sequences - Diagrams that 'commute' - Example

    I am an amateur but i find your question interesting. I thought Z/6Z = {0 , 1 , 2 , 3 , 4 , 5} ?

    That would make 3Z/6Z = {0 , 3} ?

    So (Z/6Z)/(3Z/6Z) = {0, 1, 2, 3, 4, 5}/{0, 3} = {1 , 2 , 4 , 5} <---- really not sure about this but it appears the parenthesis force the division in the center to become a SUBTRACTION of sorts ... {0, 1, 2 , 3, 4 , 5} - {0 , 3} = {1 , 2 , 4 , 5} where the elements 0 and 3 have been subtracted from Z/6Z.

    I apologize if this is way off.

    P.S. My Google Docs Viewer can't decode your pdf attachment.

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    Re: Exact Sequences - Diagrams that 'commute' - Example

    Hi Bernhard,
    I hope the following attachment helps.
    Exact Sequences - Diagrams that 'commute' - Example-mhfrings7.png
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