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Math Help - Generators of Groups

  1. #1
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    Generators of Groups

    a) List all the cyclic subgroups of <Z10, +> (Z10 = integers mod 10)

    b) Show that Z10 is generated by 2 and 5

    c) Show that Z2 x Z3 is a cyclic group

    Any help is appreciated.... Thanks.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jzellt View Post
    a) List all the cyclic subgroups of <Z10, +> (Z10 = integers mod 10)
    Take every element and work out the subgroup generated by them. For example, <5> = \{5, 0\} as 5+5=0.

    Quote Originally Posted by jzellt View Post
    b) Show that Z10 is generated by 2 and 5
    Z_{10} = <1>. That is to say, it is generated by the element 1. So, if you can prove that 1 \in <2, 5> then you are done. You can prove that 1 \in <2, 5> by noting that 2 and 5 are coprime, so...

    Quote Originally Posted by jzellt View Post
    c) Show that Z2 x Z3 is a cyclic group
    You want to find a single element which generates this group. So, try a few...
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