What is a Group? | Abstract Algebra

Welcome to group theory! In today's lesson we'll be going over the definition of a group. We'll see the four group axioms in action with some examples, and some non-examples as well which violate the axioms and are thus not groups.

In a fundamental way, groups are structures built from symmetries. We'll see this in action in the lesson, taking a look at the "Dihedral Group" of order 8, which is a group built from symmetries of a square.

More specifically, a group is a set G, together with a binary operation *, satisfying the four group axioms, which are as follows:

1. [CLOSURE] For all x, y in G, x*y is in G as well. This means G is closed under the operation *. (For example, the addition of any two integers is also an integer)

2. [ASSOCIATIVITY] For all x, y, and z in G, (x*y)*z = x*(y*z). This means G is associative under the operation *. (For example, the integers are associative under addition)

3. [IDENTITY] There exists an element e in G, such that for all x in G, e*x = x*e = x. In other words, combining any element with e in any order leaves the element unchanged. This element e is called the identity of the group because it preserves the identity of any element in combines with. (For example, the identity of the integers under addition is 0)

4. [INVERSES] For every a in G, there exists b in G such that a*b = b*a = e, in which case b is called the inverse of a, and a is the inverse of b. Notice that every element must have an inverse. (For example, the inverse of any integer under addition is its negative, like the inverse of 3 is -3 because 3 + -3 = 0)

Lesson on binary operations:    • What are Binary Operations? | Abstrac...  
*Note that some definitions of binary operation, including the one in my lesson, include that the operation must be closed. Under this definition, it is technically redundant to say a group must also be closed - since the group is surely closed by definition of binary operation. However, closure is typically listed as a group axiom regardless and is convenient to consider as a necessary feature of the group, rather than a semantic requirement for an operation to be called "binary".

If you need it, here is an introduction to set theory, and I have numerous other set theory lessons as well:    • An Introduction to Sets (Set Theory)  

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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Vallow Bandcamp: https://vallow.bandcamp.com/
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