Hello and welcome to my ongoing video course about Algebra, already consisting of 24 videos. Here, you’ll find my ongoing video series on Algebra, presented in the correct order. Alongside the videos, I provide helpful text explanations. To test your knowledge, take the quizzes and refer to the PDF version of each lesson if needed. If you have any questions about the topic, feel free to ask in the community forum. Now, without further ado, let’s get started!
Part 1 - Introduction
Algebra is a video series I started for everyone who is interested in the basics of algebra as a mathematical field. It generalises a lot of ideas from my Start Learning Mathematics series and puts them into a more abstract setting. This is not only interesting from a pure mathematical point of view but we also find a lot of applications for these concepts. The power of describing them all under a single definition is what makes algebra so useful for a lot of different parts in mathematics.
With this you now know the topics that we will discuss in this series. Some important bullet points are groups, rings, fields and vector spaces. In order to describe these things, we need to generalise a lot from the constructions of number sets. Let’s immediately start with most basic structure.
Part 2 - Semigroups
In this video, we start with our first algebraic object, which is a binary operation. For this, we only need a set and inside we can combine two chosen elements. After this, we can make it more specific and define so-called semi-groups. They act as a preview to the more powerful object group, we will discuss later.
Part 3 - Identities and Inverses
After learning what a semigroup is, we can start to disguish special elements in it. The first one is a so-called neutral element.
Part 4 - Groups
Now we start with the definition of a group. We already know that they should contain an identity and inverses. However, it turns out that one can reduce the requirements without losing these properties. We present the nice and short proofs, which also show some standard calculation techniques in groups.
Content of the video:
00:00 Introduction
00:55 Semigroup with inverses
01:42 First definition of a group
03:16 Second definition of a group
05:00 Proof for equivalence of definitions
10:10 Credits
Part 5 - Examples for Groups
Let’s look at some examples for groups. However, before we do that we should prove that inverses are uniquely given if they exist. With this, we are also able to consider a very general example of a group that is constructed from a semigroup.
Part 6 - Cancellation Property
Since we have all the inverses in a group, it’s really easy to solve equations in a group: one just have to multiply with the inverses. This simple procedure is usually known as the cancellation property of the group. We will show that if we have this property in a semigroup, then this semigroup is actually a group.
Part 7 - Abelian Groups
In a lot of examples of group, which we have discussed, the binary operation satisfies an additional property: commutativity. In general, this is not the case such that these groups get a special treatment and are called abelian groups. Let’s look at more examples and also at non-abelian groups like the symmetric group $ S_3 $.
Content of the video:
00:00 Intro
00:40 Symmetries of a triangle as a group
00:40 Symmetries of a triangle as a group
03:39 Symmetry group as set of permutations
06:05 Example of non-commutativity
07:49 Definition: abelian groups
07:49 Examples: numbers form abelian groups
09:25 Group of order 3 is abelian
13:00 Correction: The second case can actually not happen for a group with only 3 elements.
13:26 How many elements are needed to be non-abelian?
14:55 Credits
Part 8 - Integers Modulo m ⤳ Abelian Group
In this video, we will just discuss an important example, namely the modulo calculation in the integers. With this, we can define an abelian group, usually denoted by $ \mathbb{Z}/m \mathbb{Z} $.
Part 9 - Group Homomorphisms
In Linear Algebra, the structure-preserving maps are given by linear maps. A similar thing we have in group theory and we call them group homomorphisms. It turns out that single equation already fixes all the properties we need for that: $ \varphi(a \circ b) = \varphi(a) \ast \varphi(b) $.
Part 10 - Subgroups
By having large groups, it’s also possible to find subsets in this group that form groups by itself. The example that comes in mind is $(\mathbb{Z}, +)$ which is embedded in the larger group $(\mathbb{R}, +)$. Now, we will see how we can formulate this in general and which properties a subset has to fulfill such that is a well-defined subgroup.
Part 11 - Klein Four-Group
In this video, we will prove a statement for subgroups which holds for groups of finite order. We apply this result to a nice example, called the Klein Four-Group.
Part 12 - Subgroups under Homomorphisms
Let’s go back to group homomorphisms. We already know that they conserve the group properties and, therefore, we should be able to show that they also conserve subgroups. Indeed, images and preimages of subgroups under homomorphisms are subgroups as well.
Part 13 - Conjugate Subgroups
We can use the fact that images of subgroups under homomorphisms are subgroups again. Hence, if we consider a homomorphism from a given group $G$ to itself, a so-called endomorphism, then we can use a subgroup to find other subgroups. It turns out that inner automorphisms, which are endomorphisms and isomorphisms at the same time, can define an equivalence relation on the set of all subgroups of $G$.
Part 14 - Cyclic Groups
We will discuss how we can generate subgroups from a given set of elements. Some subgroups and groups can be generated by just one element, which makes them quite easy to study. They are usually called cyclic groups.
Part 15 - Examples of Cyclic Groups
We already know that cyclic groups are not so complicated because such a group can be generated by a single element. This makes a cyclic group immediately into an abelian group. Let’s see if we can write down some examples.
Part 16 - Subgroups of Cyclic Groups
In the next video, we will show that subgroups of cyclic groups are cyclic as well. For this we will use the Euclidean division for integers, also known as division with remainder.
Part 17 - Order of Group Elements
We already defined the order of a group as the numbers of elements in the group. This notion can easily be extended to single elements of the group by using the procedure of subgroup generation.
Part 18 - Left and Right Cosets
You might remember the conjugate subgroups that we can form by $gUg^{-1}$ for every subgroup $U$. These put similar subgroups into an equivalence class. However, this construction does not do anything for abelian groups. So in order to get more general results, we look at so called left cosets $g U$ and right cosets $U g$. Usually these don’t form subgroups anymore, but these are sets that help to analyze the whole group. We also get a corresponding equivalence relation.
Part 19 - Lagrange’s Theorem
By using the right cosets (or equivalently the left cosets) from the last video, we can define the number $|G:U|$. We call it the index of a subgroup $H$ in a group $G$ and it’s simply the cardinality of the set of right cosets. This leads directly to a famous formula known as Lagrange’s Theorem, named after the French mathematician Joseph-Louis Lagrange.
Content of the video:
00:00 Introduction
00:44 Right cosets
01:28 Cardinality of the collection of right cosets
03:18 Correction: disjoint means equal to empty set!
03:31 Definition: index of subgroup
04:25 Examples: index of subgroup
06:47 Lagrange’s Theorem in group theory
07:47 Lagrange’s Theorem for cyclic subgroups
08:51 Proof of Lagrange’s Theorem
10:15 Credits
Part 20 - Fermat’s Little Theorem
Since Lagrange’s Theorem says that the order of a subgroup has to be a divisor of the order of the group, it tells us something about number theory. Indeed, we can look at the integers modulo $p$, where $p$ is prime number. By definition this an additive group, but we can also define a multiplication on it. The result is a group denoted by $(\mathbb{Z} / p \mathbb{Z} )^{\times}$. By using Bézout’s identity, we can prove Fermat’s Little Theorem from number theory.
Part 21 - Normal Subgroups
We have already discussed the concept of conjugate subgroups. This concepts defines an equivalence relation on the set of subgroups of a given group $G$. Looking at an equivalence class, one can find a lot of subgroups, but, as we will see soon, the best case is that the equivalence only contains one subgroup. Such a subgroup is then called normal or self-conjugate. For the definition normal subgroups, there are a lot of equivalent statements out there and we will prove the equivalence here.
Other videos related to this topic:
Part 22 - Quotient Group
The big advantage of a normal subgroup $U \subseteq G$ is that we easily define a new group $G/U$, read as $G$ mod $U$. It’s called the quotient group or factor group and it just consists of the left cosets $gU$ for every $g \in G$. Since $U$ is a normal subgroup, the group operation easily extends to the left cosets as well. Let’s prove this and discuss some examples.
Part 23 - First Isomorphism Theorem
After defining the factor group $G/U$ for a normal subgroup $U$, we find a relation that is often now as the fundamental theorem on homomorphisms. The name already already suggest that it is about a group homomorphism $\varphi: G \rightarrow H$. We can show that the kernel $\mathrm{Ker}(\varphi)$ is already normal subgroup. So we can always divide it out to get the factor group. We can show that this one is isomorphic to the image or range $\mathrm{Ran}(\varphi)$.
Part 24 - Second Isomorphism Theorem (Diamond)
An important diclaimer upfront: the numbering of the isomorphism theorem in the literature is all over the place. Some authors might use the same enumeration as I do it here but different names and orderings are also common. For example, this second theorem is also known by the name diamond isomorphism theorem. This is simply because that one can order the relevant subgroups into a diagram in the shap of a parallelogram or diamond. Then one can form quotient groups on two different edges and finds out that they are isomorphic groups.
Summary of the course Algebra
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You can download the whole PDF here and the whole dark PDF.
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You can download the whole printable PDF here.
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Test your knowledge in a full quiz.
- Ask your questions in the community forum about Algebra.
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