Permutation Groups and Symmetric Groups | Abstract Algebra

TL;DR

Permutations are bijections that shuffle the elements of a set, and when combined through function composition, they form groups.

Transcript

let's go over what a group of permutations is will cover definitions why these things are groups we'll also cover symmetric groups and we'll see how we combine permutations in a group this will give us a whole array of new examples of groups to study as we continue learning abstract algebra now what is a permutation you may just think of it as a re... Read More

Key Insights

• 🌇 Permutations in abstract algebra are bijections that represent reordering of elements in a set.
• ❓ The operation between permutations is function composition, which is associative.
• 👥 Permutations have properties such as closure, an identity element, and inverses, making them form groups.
• 🌇 Symmetric groups represent the set of all permutations of a given set or the set of all permutations of the first n positive integers.

Questions & Answers

Q: What is a permutation and how is it defined in abstract algebra?

In abstract algebra, a permutation is a bijection from a set to itself, representing a reordering of the elements.

Q: What operation is used between permutations in abstract algebra?

The operation between permutations is function composition, where one permutation is applied first, followed by another.

Q: What properties do permutations have in abstract algebra?

Permutations have properties like associativity, closure, an identity element, and inverses, which make them form groups.

Q: How does function composition work with permutations?

Function composition of permutations is associative, meaning that composing multiple permutations in a row will yield the same result regardless of the order in which they are composed.

Q: What is the identity element in the context of composing permutations?

The identity element is the permutation that does not change any elements of the set, and composing it with any other permutation will yield the original permutation.

Q: Can permutations be composed with their inverses?

Yes, permutations can be composed with their inverses, and the result will be the identity element, which represents no change to the elements of the set.

Q: What are symmetric groups in abstract algebra?

Symmetric groups refer to the set of all permutations of a given set or the set of all permutations of the first n positive integers. They are denoted as S(a) and S(n).

Q: Can a group of permutations be a subgroup of a larger symmetric group?

Yes, a group of permutations can be a subgroup of a larger symmetric group, meaning that it is a subset of the permutations of the set, but it may not include all possible permutations.

Key Insights:

• Permutations in abstract algebra are bijections that represent reordering of elements in a set.
• The operation between permutations is function composition, which is associative.
• Permutations have properties such as closure, an identity element, and inverses, making them form groups.
• Symmetric groups represent the set of all permutations of a given set or the set of all permutations of the first n positive integers.
• A group of permutations can be a subgroup of a larger symmetric group.

Summary & Key Takeaways

• Permutations are bijections from a set to itself, representing a reordering of the elements.

• The operation between permutations is function composition, where one permutation is applied first, then another.

• Permutations have properties such as associativity, closure, an identity element, and inverses, making them form groups.