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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- Mar 20th 2008, 03:07 PMhzhang610help on abstract algebra
For which n,n≥2, do the cycles in S_{n} form a subgroup? Do the odd permutations form a subgroup?

- Mar 20th 2008, 05:56 PMThePerfectHacker
- Mar 20th 2008, 07:20 PMhzhang610
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? - Mar 20th 2008, 07:30 PMThePerfectHacker
For $\displaystyle n=2,3$ it is true. Because any permutation is actually a cycle. But if $\displaystyle n\geq 4$ then $\displaystyle (12)$ and $\displaystyle (34)$ are cyclic permutations while $\displaystyle (12)(34)$ is not a cycle.

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Is the product of two odd permutations an odd permutation?

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

Does my answer right?