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Math Help - prove that Sn is the maximal subgroup of Sn+1.

  1. #1
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    prove that Sn is the maximal subgroup of Sn+1.

    Just like the title says, this is probably quite elementary but I couldn't quite get started.
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  2. #2
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    Quote Originally Posted by ChiliNBeans View Post
    Just like the title says, this is probably quite elementary but I couldn't quite get started.

    Not "the" but "one" maximal subgroup, since there are others, say the alternating subgroup A_{n+1}.

    Now, S_n as subset of S_{n+1} can be identified as the set (in fact subgroup) of all permutations which leave one number

    out of \{1,2,...,n\} fixed, so if S_n< N\leq S_{n+1} , then N must move all the numbers in \{1,2,...,n\} . Try to use this to show that then N=S_{n+1}

    Tonio
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