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karush
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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:In $S_3$, find elements α and β such that |α| = 2,|β| = 2, and |αβ| = 3
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?karush said:here is the example I think we are supposed to follow
but...
(123)(123)=?
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.karush said:so then
$$|\alpha\beta|=(12)(23)=3$$
?
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.
"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.
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.
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.
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.