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)

*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".

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

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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.

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+WRATH OF MATH+


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[WrathofMath]

*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".

[nonentity]

This is some top-notch educational content. It deserves much better attention, imho.

[hmp]

finally, the video that helped me understand this specific lesson!! thank you so much!!!! i hope you can give practices in identifying whether the set given is a group or not :) thank you

[MrCoreyTexas]

good example in the second half, it points out that sets don't have to just be integers or real numbers / complex numbers. I've noticed this about your channel, it distinguishes itself with non conventional examples.

[raghavrathi]

Yo you are so good!!! Way better than my college teacher, who doesn't even know how to speak English properly

[arinalikes]

Thank u for sharing all this for free man, ur the GOAT

[chinaechetam]

Can you talk about partitions ? Great video by the way . I love your videos.

[miajia]

Thank you for the wonderful video I was so struggling with it before now Im good to go. By the way will you talk about field and rings after this video?

[tirtahadith]

Im a physics student, and i've just started to deeplearn about math. Wish me luck 🤞

[slaiff.]

Thanks for this video, really helped! My professor just does the problems with little explanation/teaching...😭

[samtux]

Wow, I've guessed it. First two videos were primers for a group theory.

[ericoduroboateng]

please make a video on GL(2, F) and SL(2, F)
also matrix by proving is a group

[JanaSafa-hp]

is inverse group another word for symmetric element?

[daltonkirtzinger]

are groups the same as fields? It appears so but my prof uses different notation and everything

[JTan-fqvy]

Can an element of a group has two different inverse? Is the inverse of an element in a group unique?

[jomilvallespin]

Do you have the next video about this sir?

[daleepkumar]

Sir ph no snd krna apna apse question niklbana hai ek