A Comprehensive Guide to Abstract Algebra

Unrevelling The Mystery

Motivation of Studying Abstract Algebra

Nurturing The Past

What is abstract algebra? based on the name alone, you might think it’s a similar algebra course most people take in high school, just a little more abstract. But if you open a book on abstract algebra, you’ll be in for quite a shock. It looks nothing like the algebra most people know about. So, to help you understand the subject, let’s go back in time. The year was 1800, and for some time now, people have known how to solve linear equations, quadratic equations, cubic equations, and even quartic equations. But what about equations of higher degree? Degree five, six, seven, and beyond?

Discovery of Group

A young teenager named Évariste Galois answered this question. And to do so, he used a tool that he called a “group”. Around this time, Carl Friedrich Gauss was busy making discoveries of his own. He ironed out a new technique called modular arithmetic which helped him solve many problems in number theory. Modular arithmetic shared many similarities to the groups used by Galois. The 1800s also saw a revolution in geometry. For more than 2,000 years, Euclid dominated the scene with his book The Elements, but mathematicians began to realize there are other geometries beyond the one devised by the ancient Greeks. It didn’t take long before groups were found to be a useful tool in studying these new geometries. It soon became clear that groups were a powerful tool that could be used in many different ways. So, it made sense to “abstract” out the common features of this tool used by Galois, Gauss, and others into a general tool, and then learn everything about it. Thus Group Theory was born.

And if “groups” were so useful, it’s natural to ask: would this approach work elsewhere?

Evolution of Abstract Algebra

Soon, new abstract objects began to take shape: rings, fields, vector spaces, and modules this didn’t happen overnight. It took years of hard work to find the right definitions. Too specific, and they wouldn’t be very useful. Too general, and they would be kind of boring and not very useful. But eventually, the right definitions were identified. Altogether, they form the subject we now call Abstract Algebra. At first glance, abstract algebra may not seem very applicable to the world around us. But it’s a young subject, and its usefulness continues to grow. Every year, new uses of abstract algebra are found, and not just in mathematics. Physics, chemistry, computer science, and other areas are discovering just how useful abstract algebra can be.

Note

Quick note - ‘abstract algebra’ is sometimes called ‘modern algebra’. And if you are ever at a cocktail party with mathematicians, they will simply call it “algebra”.

So, when are you ready to begin learning abstract algebra? First, you really need to know the more familiar algebra, but the most important requirements are mathematical experience and mental maturity. Have you seen many mathematical proofs before? Are you able to think very abstractly? If so, then get ready, abstract algebra will challenge you like never before.

Group

Motivation of Group

There are many different areas of mathematics: geometry, number theory, topology, and algebra and the problems in these fields can be very different from one another. But in the late 1800s, people noticed they were using the same tool to solve many different kinds of problems. This sparked an idea: instead of reinventing the wheel, why not lift the tool out of the different branches of math and study the full power of the tool in a general form? This led to the creation of a new subject called abstract algebra, and the star of this field is a structure called the ‘Group’.

Group From Different Point of View

Before we give the definitions of a group, let’s see few examples. This will highlight the wide variety of groups and help we appreciate why it took a while for mathematicians to recognize that they were using the same tool in completely different settings.

Clock Arithmetic or Modular Arithmetic

To begin, imagine you live on an alien planet where each day in $7$ hours long. Here, a clock has $7$ hours. You might be tempted to number them $1$ to $7$, but if computer science taught anything is that you should begin counting with $0$. So, we’ll number the hours $0$ to $6$. As far as the clock is concerned, there are just $7$ numbers i.e., $0,1,2,3,4,5,6.$ You can add and substract any two of these numbers. For example what is $3+5=?$

If the hour hand starts at $3$ and you add $5$ more hours, you end up at $1$. So, on this clock $3+5=1.$ Similarly, $4+3=0.$ Because, if you start at $4$ and add $3$ more hours you end up at the top which is 0. You can subtract as well. To find $1-3$ we start at $1$ and move counterclockwise for $3$ hours. You end up at $5,$ so $1-3=5.$ Clock arithmetic is also called modular arithmetic. In the example we just saw, there are $7$ hours, so we call them ‘$\text{Integers}mod7$’, mod is short for modular. At first glance, you might think this is just an amusing number game - but it has many applications in mathematics and computer science.

Symmetries of a triangle

For our next example let’s look at an equilateral triangle. We’ll number the three corners $1,2,3$

Here is the puzzle: How many different ways can you rotate & flip the triangle so it looks the same before and after?

i.e., how many different ways are there that you can pickup, flip, or rotate it around and then set it back down so that it is in the same starting orientation? You get the first transformation by picking up the triangle and setting it back down again unchanged. We’ll call this transformation $1.$ Let’s now find the other ways you can rotate and flip this triangle so that it overlaps itself. For starters, you can pick up the triangle and rotate it ${120}^0$clockwise. These give you $3,1,2.$ We call this transformation $r$, for rotation. If you perform two such rotations, you get $2,3,1.$ We call this rotation $r^2$, since we applied $r$ twice. And if you do three rotations, you return to the starting position i.e., $1,2,3$ again. This means $r^3=1$.

Next, if you pickup the triangle and flip it about the vertical axis you get $1,3,2$. We will call a flip about the axis $f$ for flip. If you flip it twice, you’ll return to the starting position, so $f^2=1$. You could also flip then rotate,i.e., $r\times f$ which gives $2,1,3$. And finally you can flip then perform two rotations, i.e., $r^2\times f$ which gives $3,2,1$ This covers all possible transformations. There are six altogether. Notice that there were two basic two basic transformations: a ${120}^0$rotation which we will call $r$ and flip about the vertical axis which we call $f$.

Aside from the unchanging transformation, each of the transformations is a combination of these two actions. Notice you can multiply any two transformations by first applying one, then the other. In particular, if you multiply any transformation by one, you get the same transformation. This is because one doesn’t actually change the triangle.

This is one way to study a shape in geometry: look at all the ways to transform the shape into itself. These transformations are called symmetries. Believe it or not, there is a close relationship between the of shape and clock arithmetic. Before we see this connection, let’s look at one final example: the integers under addition. Let’s look at a few interesting properties of the integers:

Integers $\mathbb{Z}$ under $+$

1. The integers, which we denote by the letters $\mathbb{Z}$ are the set of all whole numbers. $\text{Integers:}\mathbb{Z}=\{\dots ,-4,-3,-2,-1,0,1,2,3,4,\dots \}$

2. $\text{Let}x,y\text{be two integers}$

   $\text{Operations:}x+y,x-y\text{are integers}$ i.e., you can add and subtract any two integers.

3. $\text{Subtraction is adding negative:}7-11=7+(-11)=-4$ subtraction is really just adding with negative numbers. For this reason, it’s common to say there’s a single operation i.e., addition$\text{Single Operation:}+$

4. Closure: If you add any two integers, you get another integer. We say the integers are closed under addition.

5. But if you divide one integer by another, you may not get an integer and we say that Integers are not closed under division. $\text{For example;}3,5\in \mathbb{Z},\text{but}\frac{3}{5}\text{is not an integer}.$ Because you see if you divide this you’ll get a set outside of the integer.

6. Zero: There’s one number that has a unique property i.e., $0$. If add zero with any integer you’ll get the same integer. i.e.; $x+0=0+x=x$

As of now, we have seen three different examples.

• Clock arithmetic or modular arithmetic,

• Symmetries of a triangle,

• Integers $\mathbb{Z}$ under $+$.

Let’s now see what these three examples all have in common. To begin each of these examples have a set of objects which we call elements.

1. Elements: For the integer $mod7$ the set of elements are the numbers $0$ through $6$. For the triangle example, the set is not the triangle but the $6$ geometric transformations which we also call symmetries. And for the integers, the set is an infinite collection of whole numbers.

2. Operations: For each of these sets, there’s a way to combine any two elements to get another element in the set. We call this an operation, and we use a symbol to show that we are combining two elements. For the integers, the operation is additions and the symbol is a ${}^{\prime }{+}^{\prime }$ sign. For the integers $mod7$, the operation is also an addition. In modular arithmetic, we also use the plus sign, even though addition means something different on a clock. As with the integers, subtraction is redundant. Any subtraction problem can be rewritten as an addition problem.

Tip

Clock arithmetic: Subtraction can be rewritten with addition. For example, $2-4$ same as $2+3$ and for the symmetries of a triangle, it’s common to use a multiplication symbol. Yo could use a different symbol if you’d like, but people tend to reuse the addition and multiplication symbols a lot with groups.

3. Closure: It’s also important to remember that each of the three sets are closed under the operation i.e., if you pick any two elements $A$ and $B$ in the set and combine them, you get another element in the set.

4. Identity: All three structures also have something called identity element, this is an element that has no effect when combined with other elements. For the integers $mod7$ it’s $0$. If you take any number $x$ on the clock and add $0$ you get $x$. For the triangle, the identity element is the transformation $1$. This is a transformation where you pick up the triangle and drop it unchanged. If you multiply any transformation $y$ by the transformation $1$ you’ll get $y$. And for the integers, the identity elements is also $0$. For any integer $z,$ $z+0=0+z=z$.

5. Inverse: For every elements, there’s an opposite elements which we call the inverse of an element. For the integer under addition, the inverse of $3$ is $-3$; the inverse of $7$ is $-7$ and so on. And if you add an element and its inverse, you get $0.$ The identity element for the symmetries each transformation has an opposite as well.

   Tip

   a) For example, the inverse of the element $r$ that rotates ${120}^0$ clockwise is the transformation that’s double rotation which is $r^2$. This is because doing $3$ rotations makes a complete circle returning you to the starting position. In other words, if you multiply $r.r^2=r^3=1$ which is an identity element for the symmetries.

   b) For the integers $mod7$ the inverse of $3$ is $-3$ but negative $3$ is just $4$ on the clock. So, we say the inverse of $3$ is $4$ and $3+4=0(mod7)$ is the identity element. So for all three examples, each element $x$ has an inverse. If you’re using the plus sign for your operation, the inverse is written as $-x$. And if you’re using the multiplication sign for your operation, the inverse is written as $x^{-1}.$ Combining $x$ with its inverse gives you the identity element.

   There’s one final property that’s essential in the definition of a group i.e., the associative property.

6. Associativity: This just says that when combining three element, it doesn’t matter how you group them. You can start by combining the first two elements, or you can start with the last two elements. Either way, you’ll end up with the same answer. This in one of those properties people take for granted, but it’s important nonetheless. If grouping did make difference, much of mathematics would come to a screeching halt.

Formal Definition of Group

Let’s now give the general definition of group,

• Set of Elements: A Group is a set of elements $G$

• Operation: It has an operation which allows you to combine any two elements. Common symbols for the operations are $+,\times$, but speaking generally we’ll use an $\ast$(asterisk).

• Closed Under Addition: The group is closed under the operation i.e., if you combine any two elements in the group, you get another element in the group. $x,y\in G⟹x\ast y\in G$

• Inverse: $\mathrm{\exists }{x}^{-1}\mathrm{\forall }x$ i.e., for each element $x$ there exists an $x^{-1}$. This is an object that has the opposite effect of $x$ and when you combine $x$ and its inverse you get the identity element which we’ll call $e$ , mathematically , $x\ast x^{-1}=e$.

• Identity: If you combine any element $y$ with the identity element $e$, you get $y$ i.e., $y\ast e=e\ast y=y$

• Associativity: The elements obey the associative property $(a\ast b)\ast c=a\ast (b\ast c)$

This is the definition of group. It took mathematicians many years of work to identify these as the most essential properties.

Important

1. Groups are not required be commutative i.e., $x\ast y\ne y\ast x$. This rule was not included because it would exclude many important examples like the group of symmetries.

2. With the triangle example a flip followed by a rotation gives you something different than a rotation followed by flip.

3. If a group is commutative, we call it a commutative group. Another common name is an abelian group. If the group is not commutative then we say it’s a non-commutative or non-abelian group.

This definition of group is an abstract idea, but what do you expect from abstract algebra? Because groups are abstract itself, they generalize a lot of different things from arithmetic, algebra, geometry an more. This generality is what makes groups so powerful. The mathematics you learn in abstract algebra can be applied to many different subjects.

Arithmetic

To get things started, let’s think back to arithmetic. When studying arithmetic, you learn how to add, subtract, multiply, and divide different kinds of numbers: integer, fractions, real numbers, complex number. Then one day, probably when you least expected it, you learned that subtraction is addition in disguise!

Note

For example, $7-4$ is the same thing as $7+(-4)$. So, instead of subtracting, you’re really adding negative number. Similarly, division is multiplication disguise! $9÷5$ is the same thing as $9\times \frac{1}{5}$.

So, instead of dividing, you’re really multiplying by a fraction. So, in arithmetic, there are two operations additon( $+$ ) and multiplication($\times$). For addition, opposites are negative numbers and for multiplication, opposites are reciprocals.

In abstract algebra, we use the word “inverse” instead of “opposite”. If you combine a number with its inverse, you get an identity element. For addition, identity is $0$. If you add $5$ for its inverse $-5$ you get $0$. What makes $0$ unique is if you add it to any number, that number doesn’t change. It retains its identity. For multiplication, if you multiply a number by its inverse $-$ its reciprocal, you get $1$. What makes $1$ special is if you multiply it by any number, that number doesn’t change. So for both addition and multiplication, we have numbers, we have inverses, we have identity elements. Now we’re ready to take the textbook definition of group.

Textbook Definition of Group

• A group is a set of elements.

• The group has one operation $\ast$

• Closed under $\ast$ (Closed under operation)

• Inverses

• Identity: $e$

• Associativity Property

Remarks

1. Notice we’ve claimed “elements” instead of “numbers”, because ‘Element’ is more abstract.

2. If you combine two elements in the group, the results is also in the group.

3. Each element has an inverse.

4. If you combine an element with its inverse, you get an element we call an identity element.

5. The associativity property is a very important property because without this property you can not solve any equations.

Note

• A group is a collection of elements.

• There’s a single operation ( $\ast$ ).

• The group is closed under the operation ( $\ast$ ).

  • If you combine two elements in the group, you’ll get a third element in the group.

• There’s an identity element ( $e$ ).

• Every element has an inverse, and combining an element with its inverse gives you the identity element.

• The elements follow an associative property.

Groups: Motivation For The Definition

After reading this formal definition one can definitely ask that why these rules and what’s so special about these properties? I would like to motivate this definition with a simple example.

Imagine that we were asked to define a group. Specifically, we want to define a group for a integers under addition. We want our definition to be as simple as possible the simpler the better. One thing we would like to be able to do is solve basic equations. If we’re going to generalize algebra, if we’re going to be abstract algebra, then we need to be able to solve equations. So let’s solve an equation $x+3=5$ and keep track of the properties we use along the way.

Solve The Equation $x+3=5$

Methodology

First we add $-3$ on both sides remember subtraction is adding negative numbers. So, we need inverses in our definition. If we simplify the right-hand side we get $2.$ The simple act of adding these two integers requires a closed operation. Let’s add this to our definition. Next, we regroup the numbers on the left-hand side. To do this we need the associative in our definition of a group. Here $3+(-3)$ is zero which is the identity element. You better believe that’s going in our definition so $x=2$

$$\begin{array}{|cc|}\hline \mathbf{\text{Equation}}& \mathbf{\text{Properties}}\\ x+3=5& \text{Integers under }+\\ \Rightarrow [x+3]+(-3)=5+(-3)& \text{Inverses}\\ \Rightarrow [x+3]+(-3)=2& \text{Closed under }+\\ \Rightarrow x+[3+(-3)]=2& \text{Associativity}\\ \Rightarrow x+0=2& \text{Identity}\\ \Rightarrow x=2& \\ \hline\end{array}$$

So it turns out that the definition of a group is the simplest definition that will let you solve a basic equation.

The Definition of a Subgroup

Suppose we have a group which we call $G$. This is a common name for groups since ‘group’ starts with the letter $G$. So, a subgroup of $G$ is a subset that is also a group. A common letter used for a subgroup is $H$.

To avoid writing words there’s a math symbol used to indicate the ${}^{\prime }H\text{is a subgroup of}G{}^{\prime }.$That’s right the less than or equal sign is used to say ${}^{\prime }H\text{is a subgroup of}G{}^{\prime }i.e.,H\le G.$ This notation suggest that ${}^{\prime }H\text{is a smaller group than}G{}^{\prime }.$

Technical Postscript

Technically $G$ is a subgroup of itself. That’s why we use the ${}^{\prime }\le {}^{\prime }$ symbol. It’s not very interesting subgroup but a subgroup just the same. If $H\ne G$ then you can simply say $H<G$ and you can verbally express as, ${}^{\prime }H\text{is a proper subgroup of}G{}^{\prime }$ since $H\subset G$.

Group Multiplication Tables

When you learned arithmetic you probably had to memorize a multiplication table. The table below shows you how to multiply any two small numbers usually the integers $1$ through $10$.

In abstract algebra, you begin to work with new types of numbers. Groups behave very differently than the numbers in arithmetic. So when you’re first starting out in abstract algebra, it’s helpful to go back to the basics and make a group multiplication table.

Cayley Tables

Group Multiplication table Sometimes call Cayley Tables in honor of the British mathematician Arthur Cayley because he was the first to use them in a math paper. They worked just like the multiplication tables from arithmetic. Let’s see a simple example & then look at how Cayley Table can be used to explore the small groups.

$$\text{Group:}\times ,\{1,-1,i,-i\}$$

$\times$	$1$	$-1$	$i$	$-i$
$1$	$1$	$-1$	$i$	$-i$
$-1$	$-1$	$1$	$-i$	$i$
$i$	$i$	$-i$	$-1$	$1$
$-i$	$-i$	$i$	$1$	$-1$

Consider group under multiplication consisting of these four element $\{1,-1,i,-i\}$ it’s traditional to put the group operation in the upper left corner. Next, list the element in the same order in the header row & header column starting with the identity element. $1$ is the identity element under multiplication. So, any number multiplied by one is unchanged. This allows us to quickly fill in the first row & first column. Now, multiply and fill in the rest of the table.

Here are the important features of Cayley Tables

Important

Identity Element in the First Row and Column

• The first row and the first column of a Cayley Table always repeat the elements in the headers.

• Why? This happens because multiplying any element by the identity element leaves the element unchanged.

  • For example, in a group, if the identity is denoted as $e$, then $a\ast e=a$ and $e\ast a=a$.

Presence of the Identity Element in Every Row and Column

• The identity element appears exactly once in every row and every column of the table.

• Why? Every element in a group has an inverse. The inverse of an element $x$ is the element $y$ such that $x\ast y=e$, where $e$ is the identity element. For example, in the group of complex numbers, the inverse of $i$ is $-i$ because $i\ast -i=1$.

Symmetry About the Diagonal (For Abelian Groups)

• The table is symmetric along its diagonal. If you flip the table along the diagonal, it remains the same.

• Why? This is because the group is abelian, meaning the operation is commutative. For any two elements aaa and bbb in the group, $a\ast b=b\ast a$

Asymmetry in Non-Abelian Groups

• Non-abelian groups are not symmetric about the diagonal.

• Why? In non-abelian groups, the operation is not commutative. This means $a\ast b\ne b\ast a$ in general, so the Cayley Table is not symmetric.

No Duplicate Elements in Rows and Columns

• Each row and each column (excluding the headers) contains all the elements of the group exactly once, but possibly in a different order.

• Why? This property reflects the closure of the group operation and ensures that the group satisfies the axioms of being a set under a binary operation.

These features are fundamental to understanding the structure of Cayley Tables and provide insights into the underlying properties of groups, both abelian and non-abelian.

Story Proof

Proposition: $\text{There’s no duplicate elements in any row or column}$

Let’s deduce an outline of this proof then we’ll write the proof rigorously.

Figure 1

Figure 2

• Suppose we’re replacing the multiplication table (fig-1) with one for an arbitrary finite group (fig-2).

• Let’s assume there’s a row with duplicate elements.

• Let’s name the row $a$ and the columns $x$ & $y.$

• Then we have $a\ast x=a\ast y$.if we multiply on the left by the inverse of a you’ll get $x=y.$

  This contradiction leads us to our assumption is wrong. So,

  $$\boxed{{\displaystyle \text{There’s no row with duplicate element}}}$$

Warning

We’re not counting the headers each row & column contain all the group elements in some order. Here, we proved this happens for every group.

Rigorous Proof

Claim: $\text{In the Cayley table of a group}$ $G$,$\text{no row can contain duplicate elements.}$

Assumption: Suppose there exists a row in the Cayley table that contains duplicate elements.

Theoretical Argument: Let the row correspond to an element $x\in G$. $\therefore \mathrm{\exists }y,z\in G$

$$x\ast y=x\ast z$$

$\text{Since}$ $G$ $\text{is a group, every element has an inverse, and the group operation is associative.}$

$$⟹x^{-1}\ast (x\ast y)=x^{-1}\ast (x\ast z)$$

$$⟹(x^{-1}\ast x)\ast y=(x^{-1}\ast x)\ast z\text{ [Associativity] }$$

$$⟹e\ast y=e\ast z[\because (x^{-1}\ast x)=e]$$

$$⟹y=z$$
This contradicts our assumption that $y$ and $z$ are distinct elements. Hence, the assumption that a row in the Cayley table contains duplicate elements is false. Therefore,

$$\boxed{{\displaystyle \text{Each row in the Cayley table contains all elements of the group G exactly once, and no duplicates are present.}}}$$

Tip

We can use the same reasoning to show that there cannot be a column with duplicate elements either.

Story Proof

Groups $G$ of order $1$

We’re now going to use Cayley Tables to find the first few small groups. We’ll begin with the simplest case, groups $G$ of order 1.

Note

” Order” of a group = Number of elements in the group. & it’s written like, $|G|=\text{Order of}G$

$\ast$	$e$
$e$	$e$

Start with every group must have an identity element. We’ll use $e$ for the identity element. So, $e\ast e=e$ since it’s the only element around & it’s the identity element. So, there’s a single group of $\text{Order }1$. This cute little group is called Trivial Group.

$\ast$	$e$	$a$
$e$	$e$	$a$
$a$	$a$	$e$

Groups of $\text{Order }2$.

Now, let’s use the multiplication table to find all groups of $\text{Order }2$. We’ll call identity element $e$ & the second element $a$. Since, $e$ is the identity element $e\ast e=e$ , $e\ast a=a$ & $a\ast e=a$ this leaves only one empty square in the table. Now we can use the rule we proved earlier.

$$\boxed{{\displaystyle \text{Each row and column contains all elements. No duplicates in any row or column. }}}$$

As, every row & every column & every column must contain all the group elements there are no duplicates. Since the second row already contains $a$ this means the last cell must be $e$. So, there’s only one group of $\text{Order }2$. It turns out that this is the same group as the $\cong \mathbb{Z}/2\mathbb{Z}(\text{integers mod 2}).$ Try to convince yourself by making the Cayley Tables for the $\text{integers mod 2}$ under addition. Then comparing the two tables.

Groups of $\text{Order }3$

Now, let’s find the all groups of $\text{Order }3$. We’ll call the $3$ elements $e,a,b.$ As, $e$ is the identity element we

$\ast$	$e$	$a$	$b$
$e$	$e$	$a$	$b$
$a$	$a$
$b$	$b$

can quickly fill in the first row & first column. This leaves $4$ cells empty in our Cayley Tables.

Let’s try to riddle out what goes in the above empty cells.

It can be $a$ because this $3^{\text{rd}}$row $2^{\text{nd}}$ column already contains $a$. This means it’s either $e$ or $b$.

Caution

What happen if our guess is $e$ ?

If $a\ast a=e,$ then the final cell must be $b$, as each column has to contain all elements. But the moment we write $b$ we see a problem. The $4^{\text{th}}$ column contains two $b$’s this is not allowed. Our guess was wrong. So, $a\ast a\text{must be equal to }2b.$

We can now fill the rest of the table. Since, we’ve already used $a$ & $b$ in $3^{\text{rd}}$ row the last cell must be $e$. Since there’s $a$ & $b$ $3^{\text{rd}}$ column the empty cell must be $e$. Since the last row already contain $b$ & $e$. So, last empty cell would be $a$.

$\ast$	$e$	$a$	$b$
$e$	$e$	$a$	$b$
$a$	$a$	$b$	$e$
$b$	$b$	$e$	$a$

So, there’s only one group of $\text{Order }3$ & this is its Cayley Tables. Because this table is symmetric about the diagonal. It’s an abelian group. In fact this group is identical to the $\cong \mathbb{Z}/3\mathbb{Z}(\text{integers mod 3 under +}).$ We say they’re isomorphic groups. Here’s one way to see this. Let’s compare both of the table with each other.

If you replace $(e,a,b)\to (0,1,2)$ the two table would be th same.

Note

Notice that filling in the Cayley Tables is kind of solving the Sudoku puzzle. Each row & column must contain all the elements of the group. There can be no duplicate elements