Permutation Groups and Symmetric Groups | Abstract Algebra

Summary

This video script delves into the concept of permutations as a fundamental aspect of abstract algebra. It defines permutations as bijections from a set to itself and explains how they form groups under function composition. The script covers the properties of group operations, such as associativity, identity, and inverses, using the symmetric group notation S_a and S_n as examples. It also provides a detailed exploration of the symmetric group S3, illustrating the composition of permutations and identifying inverses through a comprehensive table.

Takeaways

• 🔄 A permutation is a reordering of elements in a set, which can be thought of as a bijection from a set to itself.
• 🔄 The operation between permutations is function composition, where one permutation is applied before another.
• 🔄 Function composition is associative, meaning the order in which compositions are performed does not affect the result.
• 🔄 The composition of permutations is closed, meaning that composing any two permutations results in another permutation.
• 🔄 There exists an identity element in permutation groups, which is the identity function that leaves all elements unchanged.
• 🔄 Every permutation has an inverse that, when composed with the original permutation, results in the identity permutation.
• 🔄 Permutations on a set form a group, with the set of all permutations of a set denoted as S_a, and specifically for the first n positive integers as S_n.
• 🔄 Symmetric groups, denoted by S_n, represent all possible permutations of a set and can be likened to the symmetries of an object.
• 🔄 A group of permutations may refer to a subgroup of a symmetric group, not necessarily containing all permutations.
• 🔄 The number of permutations for a set of n objects is n factorial, as demonstrated by the six permutations in the symmetric group S3.
• 🔄 The inverse of a permutation can be found using a composition table, which shows the result of composing any two permutations from the group.

Q & A

What is a permutation?

A permutation is a reordering of elements in a set, and more rigorously, it is a bijection from a set to itself, meaning it is a one-to-one and onto mapping that rearranges the elements of the set without changing their quantity.

Why are permutations considered groups in abstract algebra?

Permutations are considered groups because they satisfy the four group axioms: closure (composition of any two permutations results in another permutation), associativity (the composition of permutations is associative), identity (there exists an identity permutation that leaves all elements unchanged), and inverses (every permutation has an inverse that undoes its effect).

What is the operation between permutations?

The operation between permutations is function composition, where one permutation is applied after another, resulting in a new permutation.

What is the identity element in the group of permutations?

The identity element in the group of permutations is the identity function, denoted as Epsilon, which maps each element to itself, leaving the set unchanged.

How do you find the inverse of a permutation?

The inverse of a permutation is found by reversing the effect of the original permutation such that when composed with the original permutation, the result is the identity permutation. Each element is mapped to its original position before the permutation was applied.

What is the significance of the symmetric group notation S_a?

The notation S_a stands for the symmetric group on the set 'a', which is the set of all permutations of the elements in 'a'. It is called 'symmetric' because each permutation can be seen as a symmetry of the set.

What is the difference between a symmetric group and a group of permutations?

A symmetric group, denoted as S_a or S_n, contains all possible permutations of a set. A group of permutations, on the other hand, may refer to any group G that is a subgroup of a symmetric group, and does not necessarily include all permutations.

How many permutations are there in the symmetric group S3?

There are six permutations in the symmetric group S3, as the number of permutations of 'n' objects is n factorial, and for S3, n=3, so 3! = 6.

Can you provide an example of composing two permutations from the symmetric group S3?

Sure, if we take the permutations Beta and Gamma from S3 and compose them (Beta composed with Gamma), we first apply Gamma which swaps 1 with 2 and leaves 3 unchanged, and then apply Beta which swaps 1 with 3 and 2 with 2, resulting in a new permutation where 1 maps to 1, 2 maps to 3, and 3 maps to 2.

What does it mean for a permutation to be its own inverse?

A permutation is its own inverse if composing it with itself results in the identity permutation. This means that applying the permutation and then applying it again leaves the set unchanged, effectively undoing the first application.

How can you verify that the composition of two permutations results in another permutation?

You can verify this by applying the first permutation to the set, then applying the second permutation to the result of the first. If the final arrangement of elements is a bijection from the set to itself, then the composition is indeed another permutation.