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Math Help - Definition of a generator of a cyclic group?

  1. #1
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    Definition of a generator of a cyclic group?

    My book defines a generator a of a cyclic group as:

    <a> = \left \{ a^n | n \in \mathbb{Z} \right \}

    Almost immediately after, it gives an example with Z_{18}, and the generator <2>. but it says...

    <2> = \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\}

    Where are those numbers (0, 2, 4, ..., 16) coming from? If I follow the definition of a generator exactly,

    <2> = \left \{ 2^n | n \in \mathbb{Z} \right \} = \left \{ \right 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 \}= \left \{1,2,4,8,16,14,10 \right \}

    And obviously,
    \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\} \neq \left \{1,2,4,8,16,14,10 \right \}

    What?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Definition of a generator of a cyclic group?

    Quote Originally Posted by tangibleLime View Post
    My book defines a generator a of a cyclic group as:

    <a> = \left \{ a^n | n \in \mathbb{Z} \right \}

    Almost immediately after, it gives an example with Z_{18}, and the generator <2>. but it says...

    <2> = \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\}

    Where are those numbers (0, 2, 4, ..., 16) coming from? If I follow the definition of a generator exactly,

    <2> = \left \{ 2^n | n \in \mathbb{Z} \right \} = \left \{ \right 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 \}= \left \{1,2,4,8,16,14,10 \right \}

    And obviously,
    \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\} \neq \left \{1,2,4,8,16,14,10 \right \}

    What?
    In Z_{18}, the group operation is addition, not multiplication. That said, 2 is not a generator of Z_{18} because gcd(2,\ 18)\neq 1.
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  3. #3
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    Re: Definition of a generator of a cyclic group?

    Grr, I keep typing responses and the forum erases them when I hit reply...??

    Anyways,

    So how do I know it's addition? Is it just standard and assumed that with \mathbb{Z}, addition is always used for cyclic groups? What about Q, C, R, etc?

    So when it's addition, the definition of a generator changes from

    <a> = \left \{ a^n | n \in \mathbb{Z} \right \}

    to

    <a> = \left \{ n \cdot a | n \in \mathbb{Z} \right \}?
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Definition of a generator of a cyclic group?

    Quote Originally Posted by tangibleLime View Post
    So how do I know it's addition? Is it just standard and assumed that with \mathbb{Z}, addition is always used for cyclic groups? What about Q, C, R, etc?
    Yes, it is addition unless specified otherwise.

    So when it's addition, the definition of a generator changes from

    <a> = \left \{ a^n | n \in \mathbb{Z} \right \}

    to

    <a> = \left \{ n \cdot a | n \in \mathbb{Z} \right \}?
    Yes.
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  5. #5
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    Re: Definition of a generator of a cyclic group?

    Grr, I keep typing responses and the forum erases them when I hit reply...??

    Anyways,

    So how do I know it's addition? Is it just standard and assumed that with \mathbb{Z}, addition is always used for cyclic groups? What about Q, C, R, etc?
    as other have pointed out in other threads, a set is not a group, so a multiplication must be specified.

    however, with \mathbb{Z}_n, it is usually obvious from context that addition modulo n is intended, since 0 \in \mathbb{Z}_n has no multiplicative inverse. this applies to the other sets you mentioned as well.
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  6. #6
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    Re: Definition of a generator of a cyclic group?

    Ah, now that I think about it a bit more, that does make sense. Thanks!
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