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

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Homework Help Overview

The discussion revolves around properties of permutations in the symmetric group \( S_n \) and their subgroups, specifically focusing on the parity of permutations and their relationships within the group structure.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to explore the conditions under which all or half of the permutations in a subgroup are even. There are inquiries about the implications of odd permutations and their products with elements from \( A_n \). Participants also discuss the properties of cycles and their effects on permutation parity.

Discussion Status

Some participants are actively engaging with the problems, sharing their thoughts and approaches. There is an indication that hints have been requested rather than direct solutions, and some participants are reflecting on their own mistakes and learning from them.

Contextual Notes

Participants are navigating through concepts of group theory, specifically focusing on the symmetric and alternating groups, while adhering to homework guidelines that discourage direct solutions.

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

What thoughts have you had on these problems thus far?
 
For the last one, I experimented with various sizes of [itex]\sigma[/itex]. The others I have no idea how to approach (please do not spoonfeed, just give hints).

Thanks,

Mary
 
Last edited:
(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?
 
I've solved the first two...now about the last one

NVM: i made tons of mistakes, leading to an erroneous result.
 
Last edited:
What do you know about the group generated by σ?
 

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