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Math Help - Order of an element

  1. #1
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    Order of an element

    I am a bit confused about this. Could someone guide me in the right direction? Thank you.

    What is the maximum order of an element of S5xS7?
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  2. #2
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    Re: Order of an element

    As my calculations, I think the maximum order of an element of S_{5} \times S_{7} is 60, which is the order of an element with a cycle of length 5 in the first component and a permutation which is product of a cycle of length 3 and a cycle of length 4, in the second component. One of such element is ((12345),(123)(4567)).
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  3. #3
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    Re: Order of an element

    Thank you. I am still confused. Why shouldn't be an element of order 35 = 5x7 then? Why did you split the seven into three and four?

    thanks again for your help.
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  4. #4
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    Re: Order of an element

    The order of each element (g, h) in G H is the least common multiple of the orders of g and h:
    | (g, h) | = lcm( | g |, | h | ).
    Besides, in Sn, the order of any permutation x which is a product of k disjoint cycles of finite lengths m1 ,...,mk is the least common multiple of these lengths , i.e. | x | = lcm( m1 ,...,mk ) and we have n=m1+...+mk(by considering cycles of length 1).
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  5. #5
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    Re: Order of an element

    It makes sense, thank you very much.
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