Thread: 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?

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?

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?

Originally Posted by hzhang610

Is a product of two cycles necessarily a cycle?

No. So does that mean n=2?

For it is true. Because any permutation is actually a cycle. But if then and are cyclic permutations while 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.