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Math Help - Find two generators of the group

  1. #1
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    Find two generators of the group

    Hi there,

    I'm struggling with this question: find two elements of the group G = Z2 x Z3 x Z4 that generate the group. In Z2 x Z3, which are coprime, (1,1) is a generator with order 6, but this doesn't extend then to (1,1,1) as a generator of G, because it has order 12, not 24 as required, so I can't even find one generator! Can anyone offer any advice? I know how to solve this problem if the numbers were coprime but that's not the case here.

    Thanks for any help.
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  2. #2
    Member Sylvia104's Avatar
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    Re: Find two generators of the group

    Thatís because G is not cyclic.
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  3. #3
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    Re: Find two generators of the group

    Dang!
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  4. #4
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    Re: Find two generators of the group

    in other words, we actually need to find TWO generators, to be able to describe G = <a,b>. my recommendation is a = (1,0,0) and b = (0,1,1) but a = (1,1,0) and b = (0,0,1) works equally well.

    that is, you can regard G as Z2xZ12, or as Z6xZ4.

    either way, 12 is the maximum possible order:

    12(a,b,c) = (12a,12b,12c) = (0,0,0) (since 12a = 0 (mod 2), 12b = 0 (mod 3) and 12c = 0 (mod 4).

    looking at G as Z2xZ12, we have 12(a,b) = (12a,12b) = (0,0), since 12a = 0 (mod 2) and 12b = 0 (mod 12).

    looking at G as Z6xZ4, we have 12(a,b) = (12a,12b) = (0,0), since 12a = 0 (mod 6) and 12b = 0 (mod 4).

    it turns out we do actually have elements of order 12: (0,1,1) is one such element, since lcm(3,4) = 12.

    in Z2xZ12, this would be (0,1) (identifying (1,1) in Z3xZ4 with 1 in Z12) which clearly has order 12 (since 1 has order 12 in Z12).

    in Z6xZ4, this would be (4,1) (identifying (1,1) in Z2xZ3 with 1 in Z6). it isn't so "obvious" that this has order 12, but note that the first coordinate is only 0 every 3rd multiple, and the second coordinate is only 0 every 4th multiple.
    Thanks from patrickmustard
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  5. #5
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    Re: Find two generators of the group

    Wow...thank you so much Deveno. That makes perfect sense.
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