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Math Help - External Direct Product

  1. #1
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    External Direct Product

    I need to determine the number of elements in Z_25 direct product with Z_5.

    i know that i need to count the elements with property that (lcm(|a|,|b|))= 5

    and then break in to the cases that |a|=|b| = 5. the other case will be...

    But what i am missing is know how choices can there be for both and b.

    My book says that when |a|=|b| = 5. there are 4 choices for a and four choices of b. i dont know why.

    I need help coming up with the choices for a and b. i need to know how many and what are they.
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  2. #2
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    Quote Originally Posted by Juancd08 View Post
    I need to determine the number of elements in Z_25 direct product with Z_5.
    By definition \mathbb{Z}_{25}\times \mathbb{Z}_5 = \{ (a,b) | a\in \mathbb{Z}_{25}, b\in \mathbb{Z}_5 \}. There are 25\cdot 5 = 125 such elements.

    This is Mine 11,2th Post!!!
    Last edited by ThePerfectHacker; November 8th 2008 at 02:43 PM.
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  3. #3
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    how many elements of a specific order

    I knew that, but i what I wanted to know is how many elements of a determined order is in Z_25 + Z_5. take the example how many elements of order 5. I know 5=lcm(a,b), but i dont know how to finish it.
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    Quote Originally Posted by Juancd08 View Post
    I knew that, but i what I wanted to know is how many elements of a determined order is in Z_25 + Z_5. take the example how many elements of order 5. I know 5=lcm(a,b), but i dont know how to finish it.
    If (a,b) has order 5 it means: |a|=|b|=5 or |a|=5,|b|=1 or |a|=1,|b|=5.

    That should help you solve the problem.
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  5. #5
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    sorry

    Yeah but what i need to know is when does |a|=5. same for b.

    Sorry for not asking the right question from the beggining.
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  6. #6
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    Quote Originally Posted by Juancd08 View Post
    Yeah but what i need to know is when does |a|=5. same for b.

    Sorry for not asking the right question from the beggining.
    For |a| = 5 we can have a=5,10,15,20
    For |b| = 5 we can have b=1,2,3,4
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