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Math Help - Non-abelian subgroup T of [math]S_{3} \times Z_{4}[/math]

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    Non-abelian subgroup T of [math]S_{3} \times Z_{4}[/math]

    There is a nonabelian subgroup T of S_{3} \times Z_{4} of order 12 generated by elements a, b such that  |a|= 6, a^{3}=b^{2}, ba=a^{-1}b.

    What are the generaters a, b for the above group T such that T = <a , b> and what are the elements of T?
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    Quote Originally Posted by aliceinwonderland View Post
    There is a nonabelian subgroup T of S_{3} \times Z_{4} of order 12 generated by elements a, b such that  |a|= 6, a^{3}=b^{2}, ba=a^{-1}b.

    What are the generators a, b for the above group T such that T = <a , b> and what are the elements of T?
    The order of an element (x,y)\in S_{3} \times Z_{4} is the least common multiple of the order of x in S_3 and the order of y in Z_4.

    Elements of S_3 (apart from the identity) have order 2 or 3. Elements of Z_4 have order 2 or 4. So to get an element a=(x,y)\in S_{3} \times Z_{4} to have order 6, you must choose x to have order 3 and y to have order 2.

    Next, if a has order 6 then a^{3}=b^{2} must have order 2 and so b must have order 4. So if b=(z,w) then z must have order 1 or 2 in S_3, and w must have order 4 in Z_4.

    That should be enough to get you started.
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