# 412.42 - Finding elements in S_3

• MHB
• karush
In summary, the conversation discusses finding elements α and β in $S_3$ with specific orders, and determining their product's order. It suggests listing all six elements and using trial and error. The order of a 2-cycle and 3-cycle is mentioned, and the need to specify the product (12)(13) is emphasized. The conversation ends with a comment about the topic being difficult.
karush
Gold Member
MHB
In $S_3$, find elements α and β such that |α| = 2,|β| = 2, and |αβ| = 3

I hate to be "this guy" again, but a thread title of "412.42" isn't very useful to the community. Please make sure thread titles briefly describe the question being asked. (Wave)

karush said:
In $S_3$, find elements α and β such that |α| = 2,|β| = 2, and |αβ| = 3
$S_3$ only has six elements, so you can list them all and do the question by trial and error. The elements consist of three transpositions ($(12)$, $(13)$ and $(23)$) and two 3-cycles ($(123)$ and $(132)$), the remaining element being the identity. Choose two elements with order 2, multiply them together and see whether the product has order 3.

karush said:
here is the example I think we are supposed to follow
but...

(123)(123)=?
The question is asking you to find two elements of order 2 whose product has order 3. So, what is the order of a 2-cycle and what is the order of a 3-cycle?

so then
$$|\alpha\beta|=(12)(23)=3$$
?

karush said:
so then
$$|\alpha\beta|=(12)(23)=3$$
?
If you mean $|\alpha\beta|=|(12)(23)|=3$ then you're on the right track. But you'll need to specify the product $(12)(13)$, rather than just stating that it has order 3.

ok, much mahalo,

this stuff is strange!

Last edited:

## 1. What is the purpose of "412.42 - Finding elements in S_3"?

The purpose of "412.42 - Finding elements in S_3" is to identify and locate specific elements within the mathematical group S_3, also known as the symmetric group of order 3.

## 2. How is "412.42 - Finding elements in S_3" used in mathematics?

"412.42 - Finding elements in S_3" is used in mathematics to study and analyze the properties and structure of the symmetric group S_3, which has applications in various branches of math such as group theory, abstract algebra, and combinatorics.

## 3. What methods are used in "412.42 - Finding elements in S_3"?

The methods used in "412.42 - Finding elements in S_3" involve understanding the structure and properties of the symmetric group S_3, as well as using techniques from group theory, such as Cayley tables and cycle notation, to identify and locate specific elements within S_3.

## 4. Can "412.42 - Finding elements in S_3" be applied in real-world situations?

Yes, the concepts and techniques used in "412.42 - Finding elements in S_3" can be applied in real-world situations, such as in cryptography, computer science, and chemistry, where the properties of symmetric groups are relevant.

## 5. Are there any limitations to "412.42 - Finding elements in S_3"?

Like any mathematical concept, "412.42 - Finding elements in S_3" has its limitations. It may not be applicable to other mathematical groups, and the techniques used may become more complex as the size of the group increases.

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