Definition of a Group and Examples of Groups

TL;DR

A group is a set with a binary operation that follows the properties of associativity, existence of an identity element, and existence of inverses.

Transcript

hey YouTube let's continue our discussion of abstract algebra today we're going to talk about groups so first let's look at the definition of a group so definition so a group is a set is a set G with star a binary operation on G a binary operation on G such that the following three conditions hold so the first condition is that star is an associati... Read More

Key Insights

• 😫 A group is a set with a binary operation that follows the properties of associativity, existence of an identity element, and existence of inverses.
• #️⃣ The set of integers, rational numbers, real numbers, and complex numbers under addition are examples of groups.
• #️⃣ The natural numbers under addition and the natural numbers with 0 added fail to be groups due to the lack of identity and inverse elements.
• ✖️ The rational numbers under multiplication with 0 excluded form a group, while the rational numbers under multiplication as a whole do not, as 0 lacks an inverse.

Questions & Answers

Q: What are the three conditions that a binary operation in a group must satisfy?

The binary operation must be associative, there must be an identity element that combines with any element to give that element, and for every element, there must exist an inverse element that, when combined, gives the identity element.

Q: Why is the set of natural numbers under addition not a group?

The set of natural numbers lacks an identity element, as there is no natural number that can be added to any other number and give that number. Therefore, it does not satisfy the criteria of a group.

Q: Why does adding 0 to the natural numbers still not make it a group?

Even with the addition of 0, the natural numbers fail to be a group because the nonzero numbers do not have inverse elements. In a group, every element must have an inverse, which is not the case for the nonzero natural numbers.

Q: Are there any examples of groups that are not related to numbers?

Yes, there are many examples of groups that do not involve numbers, such as groups formed by rotations, symmetries, or functions.

Summary & Key Takeaways

• A group is a set with a binary operation that satisfies the properties of associativity, existence of an identity element, and existence of inverses.

• Examples of groups include the set of integers under addition, the set of rational numbers under addition, the set of real numbers under addition, and the set of complex numbers under addition.

• The set of natural numbers under addition is not a group as it lacks an identity element, and adding 0 to the natural numbers still does not make it a group due to the lack of inverse elements.