Definition of the Symmetric Group

TL;DR

Symmetric groups are sets of bijections, forming a group under function composition.

Transcript

I wanted to make a quick video to briefly discuss symmetric groups so we have a set X so this is a set and we're going to have another set s sub X and we're going to say that this is the set of all by junctions so by jek shion's so all maps that are one-to-one and onto from X to X and this is actually a group right this is called the symmetric grou... Read More

Key Insights

• 👥 Symmetric groups consist of bijections that form a group under function composition.
• 👥 Function composition in symmetric groups is associative.
• 👥 The identity element in symmetric groups is the identity function.
• 👥 Inverses of bijections in symmetric groups are also bijections.
• 😫 Symmetric groups are often studied by restricting the set X to finite elements.
• 👥 Array notation is utilized to represent elements of symmetric groups concisely.
• ↔️ Composing elements in symmetric groups involves applying function composition from right to left.

Questions & Answers

Q: What are symmetric groups, and why are they called symmetric groups?

Symmetric groups are sets of bijections from a set to itself that form a group under function composition. They are called symmetric groups because their elements are bijections that preserve the symmetry of the set.

Q: How are symmetric groups proven to be groups?

Symmetric groups are proven to be groups by satisfying the associative property, having an identity element (the identity function), and possessing inverses. These properties ensure that the group operation is well-defined and that inverses exist for each element.

Q: What is the significance of restricting the set X in symmetric groups?

Restricting the set X to be finite allows for easier representation and computation of symmetric groups. It enables the elements of the group to be listed using numbers or symbols, simplifying the understanding of group elements.

Q: How is array notation used in symmetric groups?

Array notation is a way to represent elements of symmetric groups in a simplified format. It consists of two rows, with the top row listing elements of the set and the bottom row indicating where each element is mapped under a bijection.

Summary & Key Takeaways

• Symmetric groups consist of bijections from a set to itself, forming a group under function composition.

• The group operation is function composition, where composing two bijections results in another bijection.

• Symmetric groups are proven to be groups by satisfying the associative property, having an identity element, and possessing inverses.