# help on abstract algebra

• Mar 20th 2008, 03:07 PM
hzhang610
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?
• Mar 20th 2008, 05:56 PM
ThePerfectHacker
Quote:

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?

Quote:

Do the odd permutations form a subgroup?
Is the product of two odd permutations an odd permutation?
• Mar 20th 2008, 07:20 PM
hzhang610
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.

• Mar 20th 2008, 07:30 PM
ThePerfectHacker
Quote:

Originally Posted by hzhang610
Is a product of two cycles necessarily a cycle?

No. So does that mean n=2?

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.

Quote:

Is the product of two odd permutations an odd permutation?

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