# Thread: help on abstract algebra

1. ## 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?

2. Originally Posted by hzhang610
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?

3. 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.

4. Originally Posted by hzhang610
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.