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Abstract Algebra Problems

  1. Feb 6, 2005 #1
    Hello,

    I am a student at CMU, enrolled in the Abstract Algebra class.

    I'm having trouble with a few problems, see if you can figure them out.

    Show that for every subgroup $J$ of $S_n|n\geq 2$, where $S$ is the symmetric group, either all or exactly half of the permutations in $J$ are even.

    Consider $S_n|n\geq 2$ for a fixed $n$ and let $\sigma$ be a fixed odd permutation. Show that every odd permutation in $S_n$ is a product of $\sigma$ and some permutation in $A_n$.

    Show that if $\sigma$ is a cycle of odd length, then $\sigma^2$ is a cycle

    Thanks!

    Mary
     
  2. jcsd
  3. Feb 6, 2005 #2

    Hurkyl

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    Replace the $...$ with [ itex ]...[ /itex ] (without the spaces) to get the typesetting.

    What thoughts have you had on these problems thus far?
     
  4. Feb 6, 2005 #3
    For the last one, I experimented with various sizes of [itex]\sigma[/itex]. The others I have no idea how to approach (please do not spoonfeed, just give hints).

    Thanks,

    Mary
     
    Last edited: Feb 7, 2005
  5. Feb 7, 2005 #4

    Hurkyl

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    (note the direction of the slash on [ /itex ])

    I think the result of the middle question is a big clue to the first problem.

    What parity does the product of two odd permutations have?
     
  6. Feb 7, 2005 #5
    I've solved the first two...now about the last one

    NVM: i made tons of mistakes, leading to an erroneous result.
     
    Last edited: Feb 7, 2005
  7. Feb 7, 2005 #6

    Hurkyl

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    What do you know about the group generated by σ?
     
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