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Thread: Cyclic groups.

  1. #1
    Super Member Deadstar's Avatar
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    Cyclic groups.

    Q. To what product in standard form is $\displaystyle C_{20}$ x $\displaystyle C_{30}$ isomorphic?

    I got... $\displaystyle 20 = 2^2 \cdot 5 = 4 \cdot 5$ and $\displaystyle 30 = 2 \cdot 3 \cdot 5$

    But how do i arrange these? I imagine it should be something like $\displaystyle C_{2 \cdot ---}$ x $\displaystyle C_{3 \cdot ---}$ But im not sure where the 5's and the 4 should go.

    On a side note how do i put letters into latex so that they look like x and not $\displaystyle x$?
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  2. #2
    Super Member Gamma's Avatar
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    $\displaystyle C_{20} \cong C_4 \times C_5$
    $\displaystyle C_{30} \cong C_2 \times C_3 \times C_5$
    $\displaystyle C_{20} \times C_{30} \cong (C_4 \times C_5) \times (C_2 \times C_3 \times C_5) \cong C_4 \times C_2 \times C_3 \times C_5 \times C_5$

    Basically keep the primes together and list in increasing order of the primes, but then within each prime go from big to small I think this is standard. But there are really two methods that you can write, this is the elementry divisor one, the other is invariant factors.

    That way you group by the highest powers of all the available primes.

    $\displaystyle C_4 \times C_2 \times C_3 \times C_5 \times C_5 \cong (C_4 \times C_3 \times C_5) \times (C_2 \times C_5) \cong C_{60} \times C_{10}$

    Notice $\displaystyle C_4 \not \cong C_2 \times C_2$ so you cannot split that up from the $\displaystyle C_{20}$ one.
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  3. #3
    Super Member Gamma's Avatar
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    oh i missed the second question, if you are looking for the cross product symbol it is just \times
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