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Math Help - Direct Product

  1. #1
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    Direct Product

    Let G={1, 7, 17, 23, 49, 55, 65, 71} under multiplication modulo 96. Express G as an external and an internal direct product of cyclic groups.

    I know that lGl =8, so G is isomorphic to either Z8, or G is isomorphic to the external direct product of Z4 Z2, or G is isomorphic to the external direct product of Z2 Z2 Z2. But I'm not sure what to do next.
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  2. #2
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    Quote Originally Posted by bethh View Post
    Let G={1, 7, 17, 23, 49, 55, 65, 71} under multiplication modulo 96. Express G as an external and an internal direct product of cyclic groups.

    I know that lGl =8, so G is isomorphic to either Z8, or G is isomorphic to the external direct product of Z4 Z2, or G is isomorphic to the external direct product of Z2 Z2 Z2. But I'm not sure what to do next.

    First, find out the orders of the different elements, for instance: 7^2=49\,,\,7^3=343=55\!\!\!\!\pmod{96}\,,\,7^4=385  =1\!\!\!\!\pmod{96}\Longrightarrow\,ord(7)=4 , and this already tells you that G can NOT be the direct product \mathbb{Z}_2\times\mathbb{Z}\times\mathbb{Z}_2 . Well, do the next ones now.

    Tonio
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  3. #3
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    Patterns

    In your problem, you can also notice a pattern:

    49 = 48+1
    55 = 48+7
    65 = 48+17
    71 = 48+23

    so you might think this was { 1,7,17,23 } x { 0,1}.
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