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Thread: Direct product

  1. #1
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    Commutator subgroup

    Let $\displaystyle G_{1} \times G_{2}=Z_{12} \times S_{24}$.
    1. The commutator subgroup of $\displaystyle Z_{12} \times S_{24}$ I got is $\displaystyle e \times A_{24}$.
    2. The quotient group of $\displaystyle \frac{Z_{12} \times S_{24}}{e \times A_{24}} = Z_{12} \times Z_{2}$.

    Is above 1&2 correct?
    Since the maximum order of elements in $\displaystyle Z_{12} \times Z_{2}$ is 12, the above 2 seems wrong. ((12 * 24!) / (0.5 *24!) = 24 ).
    If 2 is wrong, what is the quotient group of 2?
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
    Let $\displaystyle G_{1} \times G_{2}=Z_{12} \times S_{24}$.
    1. The commutator subgroup of $\displaystyle Z_{12} \times S_{24}$ I got is $\displaystyle e \times A_{24}$.
    2. The quotient group of $\displaystyle \frac{Z_{12} \times S_{24}}{e \times A_{24}} = Z_{12} \times Z_{2}$.

    Is above 1&2 correct?
    Since the maximum order of elements in $\displaystyle Z_{12} \times Z_{2}$ is 12, the above 2 seems wrong. ((12 * 24!) / (0.5 *24!) = 24 ).
    If 2 is wrong, what is the quotient group of 2?
    We need to know that $\displaystyle (G_1\times G_2)' = G_1' \times G_2'$. To prove this just consider a commutator, $\displaystyle c=(a_1,a_2)(b_1,b_2)(a_1^{-1},a_2^{-1})(b_1^{-1},b_2^{-1})$ where $\displaystyle a_1,b_1\in G_1$ and $\displaystyle a_2,b_2\in G_2$. We can rewrite $\displaystyle c = (a_1b_1a_1^{-1}b_1^{-1}, a_2b_2a_2^{-1}b_2^{-1})$. But $\displaystyle a_1b_1a_1^{-1}b_1^{-1}\in G_1'$ and $\displaystyle a_2b_2a_2^{-1}b_2^{-1} \in G_2'$. Therefore, the subgroup generated by the commutators i.e. $\displaystyle (G_1\times G_2)'$ is equal to $\displaystyle G_1'\times G_2'$.

    Now the commutator subgroup of $\displaystyle \mathbb{Z}_{12}$ is $\displaystyle \{ e \}$ (not $\displaystyle e$ !) While it is an established result that the commutator of $\displaystyle S_{24}$ is $\displaystyle A_{24}$.
    Therefore, I agree with your result.
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  3. #3
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    2. is definitely correct. And note that the order of (Z/12)X(Z/2) is 24 not 12. The highest order of a single element is 12 but there are 24 distinct elements in the whole group.
    Last edited by bulls6x; Jan 8th 2009 at 03:15 PM.
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