A presentation for the symmetric group

Thread starter tgt
Start date Apr 17, 2009
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In summary, the symmetric group, denoted as S<sub>n</sub>, represents all possible permutations of a set with n elements. It can be represented using cycle notation or as a set of permutations. Some applications of the symmetric group include abstract algebra, group theory, and cryptography. It is closely related to other mathematical concepts and can be used to model real-life situations such as in Rubik's Cube.

Apr 17, 2009

tgt

Homework Statement

How to find such a presentation? Where would one start?

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Apr 17, 2009

matt grime

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Homework Helper

Well, first you need a set of elements that generates. What is the obvious candidate?

1. What is the symmetric group?

The symmetric group, denoted as S_n, is a mathematical concept that represents all possible permutations of a set with n elements. In simpler terms, it is the group of all possible ways to arrange n distinct objects in different orders.

2. How is the symmetric group represented?

The symmetric group is typically represented using cycle notation or as a set of permutations. For example, the symmetric group of order 3, denoted as S₃, can be represented as {(1), (12), (13), (23), (123), (132)}, where a number in parenthesis represents a cycle.

3. What are some applications of the symmetric group?

The symmetric group has various applications in different fields, such as in abstract algebra, group theory, and combinatorics. It is also used in cryptography, coding theory, and quantum mechanics.

4. How is the symmetric group related to other mathematical concepts?

The symmetric group is closely related to other mathematical concepts, such as permutations, groups, and symmetric polynomials. It also has connections to other areas of mathematics, such as topology and geometry.

5. Can you give an example of a real-life situation that can be modeled using the symmetric group?

One real-life application of the symmetric group is in Rubik's Cube. The different possible permutations of the cube's colored squares can be represented using the symmetric group. Each move on the cube corresponds to a permutation, and solving the cube involves finding the correct sequence of permutations to return it to its original state.