Can All Subgroup Permutations in S_n Be Even or Half Even?

[ti89fr33k]

Hello,

I am a student at CMU, enrolled in the Abstract Algebra class.

I'm having trouble with a few problems, see if you can figure them out.

Show that for every subgroup $J$ of $S_n|n\geq 2$, where $S$ is the symmetric group, either all or exactly half of the permutations in $J$ are even.

Consider $S_n|n\geq 2$ for a fixed $n$ and let $\sigma$ be a fixed odd permutation. Show that every odd permutation in $S_n$ is a product of $\sigma$ and some permutation in $A_n$.

Show that if $\sigma$ is a cycle of odd length, then $\sigma^2$ is a cycle

Thanks!

Mary

 

[Hurkyl]

Replace the $...$ with [ itex ]...[ /itex ] (without the spaces) to get the typesetting.

What thoughts have you had on these problems thus far?

 

[ti89fr33k]

For the last one, I experimented with various sizes of $\sigma$. The others I have no idea how to approach (please do not spoonfeed, just give hints).

Thanks,

Mary

 

[Hurkyl]

(note the direction of the slash on [ /itex ])

I think the result of the middle question is a big clue to the first problem.

What parity does the product of two odd permutations have?

 

[ti89fr33k]

I've solved the first two...now about the last one

NVM: i made tons of mistakes, leading to an erroneous result.

 

[Hurkyl]

What do you know about the group generated by σ?