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Math Help - Finding element of maximal order in symmetric group

  1. #1
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    Finding element of maximal order in symmetric group

    Would someone show me the method to find an element of maximal order in S_7? Is there a general technique for this?
    I think that this element could have order 12=3 \times4 since I can find a (4,3) cycle element in S_7 like (1234)(567) which has order 12.
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  2. #2
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    Quote Originally Posted by jackie View Post
    Would someone show me the method to find an element of maximal order in S_7? Is there a general technique for this?
    I think that this element could have order 12=3 \times4 since I can find a (4,3) cycle element in S_7 like (1234)(567) which has order 12.
    12 is the answer. if there exists an integer m=p_1^{k_1}p_2^{k_2} \cdots p_r^{k_r}, where p_1, \cdots, p_r are distinct primes, such that p_1^{k_1} + p_2^{k_2} + \cdots + p_r^{k_r} = n, then the maximal order of elements in S_n is m.

    for example, since 12=2^2 \times 3 and 2^2 + 3 = 7, the maximal order of elements in S_7 is 12.

    see section 4 (the prime connection) of this paper for more details. don't get scared, it's very easy to understand!
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