Youtor
What is a Group Abstract Algebra
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=dfCNJmT0-uI
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! • • ******************************************************************** • The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work. • • Vallow Bandcamp: https://vallow.bandcamp.com/ • Vallow Spotify: https://open.spotify.com/artist/0fRtu... • Vallow SoundCloud: / benwatts-3 • ******************************************************************** • • +WRATH OF MATH+ • • ◆ Support Wrath of Math on Patreon: / wrathofmathlessons • • Follow Wrath of Math on... • ● Instagram: / wrathofmathedu • ● Facebook: / wrathofmath • ● Twitter: / wrathofmathedu • • My Music Channel: / seanemusic
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