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  1. #1
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    help on abstract algebra

    For which n,n≥2, do the cycles in S_{n} form a subgroup? Do the odd permutations form a subgroup?
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  2. #2
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    Quote Originally Posted by hzhang610 View Post
    For which n,n≥2, do the cycles in S_{n} form a subgroup?
    Is a product of two cycles necessarily a cycle?

    Do the odd permutations form a subgroup?
    Is the product of two odd permutations an odd permutation?
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  3. #3
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    Is a product of two cycles necessarily a cycle?

    No. So does that mean n=2?


    Is the product of two odd permutations an odd permutation?

    No. Therefore, the odd permutations does not form a subgroup.

    Does my answer right?
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  4. #4
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    Quote Originally Posted by hzhang610 View Post
    Is a product of two cycles necessarily a cycle?

    No. So does that mean n=2?
    For n=2,3 it is true. Because any permutation is actually a cycle. But if n\geq 4 then (12) and (34) are cyclic permutations while (12)(34) is not a cycle.

    Is the product of two odd permutations an odd permutation?

    No. Therefore, the odd permutations does not form a subgroup.

    Does my answer right?
    Correct. And the reason why it is not a subgroup is because a subgroup needs to be closed.
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