Abstract Algebra Problems

In summary, the conversation is about a student seeking help with problems in Abstract Algebra, specifically related to subgroups of the symmetric group and odd permutations. The student receives hints on how to approach the problems and eventually solves them. The conversation also includes a discussion on the parity of products of odd permutations and the group generated by a specific permutation.
  • #1
ti89fr33k
13
0
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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  • #2
Replace the $...$ with [ itex ]...[ /itex ] (without the spaces) to get the typesetting.

What thoughts have you had on these problems thus far?
 
  • #3
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:
  • #4
(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?
 
  • #5
I've solved the first two...now about the last one

NVM: i made tons of mistakes, leading to an erroneous result.
 
Last edited:
  • #6
What do you know about the group generated by σ?
 

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that deals with the study of mathematical structures such as groups, rings, and fields. It focuses on the properties and relationships between these structures and their operations.

2. What are some common applications of abstract algebra?

Abstract algebra has various applications in fields such as cryptography, coding theory, computer science, physics, and engineering. It is also used in areas such as economics, statistics, and biology.

3. What are some key concepts in abstract algebra?

Some key concepts in abstract algebra include groups, rings, fields, homomorphisms, isomorphisms, and substructures. These concepts help to define and understand the properties and relationships between different mathematical structures.

4. What are some common topics covered in abstract algebra problems?

Some common topics covered in abstract algebra problems include algebraic equations, group theory, ring theory, vector spaces, and linear transformations. These topics help to develop problem-solving skills and a deeper understanding of abstract algebra concepts.

5. How can one improve their skills in solving abstract algebra problems?

Practicing regularly with different types of abstract algebra problems is the best way to improve problem-solving skills. It is also helpful to read textbooks and attend lectures or workshops to gain a better understanding of abstract algebra concepts and techniques.

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