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Math Help - Cyclic Groups - Is it a Cyclyc group?

  1. #1
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    Cyclic Groups - Is it a Cyclyc group?

    Z_{10}\times Z_{15} , while Z is the integers (I believe the action should be '+' for it to be a group, otherwise, for instance in multiplication, (0,0) has no inverse element).

    How do I generally check whether it's a cyclic group? It has 150 elements

    Thanks
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by adam63 View Post
    Z_{10}\times Z_{15} , while Z is the integers (I believe the action should be '+' for it to be a group, otherwise, for instance in multiplication, (0,0) has no inverse element).

    How do I generally check whether it's a cyclic group? It has 150 elements

    Thanks
    You need to find a generator. However, this group is NOT a cyclic group.

    Assume (a, b) generates your group. Then (a, b)^n \neq (0, 0) for all 0<n<150 (why?). However, take n=30...

    Something to think about: Why does this work? Why the number 30?
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  3. #3
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    Nice, gcd(10,15)=30.
    And, since (a,b) is the generator, then |(a,b)|=150.

    Got it ! Thank you very much!
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by adam63 View Post
    Nice, gcd(10,15)=30.
    And, since (a,b) is the generator, then |(a,b)|=150.

    Got it ! Thank you very much!
    lcm, not gcd...but yeah. Can you see the generalisation? It is a neat theorem, and one to remember...
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