Introduction to Group Theory

Group theory is the mathematical study of symmetry and structure. Instead of focusing on what particular objects are made of, group theory studies the operations that can be performed on them and the rules those operations obey.

The same abstract structure appears in many different places:

• adding integers,
• rotating a polygon,
• permuting a collection of objects,
• multiplying invertible matrices,
• and studying symmetries of equations.

Once we recognize that these examples satisfy the same axioms, we can prove results about all of them at once.

Table of contents

What is a group?

A group is a set $G$ together with an operation

that combines two elements $a,b\in G$ to produce another element $a\ast b\in G$.

The pair $(G,\ast)$ is a group when the following four properties hold.

1. Closure

For every $a,b\in G$,

The operation never takes us outside the set.

2. Associativity

For every $a,b,c\in G$,

Associativity allows us to write $a\ast b\ast c$ without ambiguity. Notice that associativity is not the same as commutativity: it changes the placement of parentheses, not the order of the elements.

3. Identity element

There exists an element $e\in G$ such that, for every $a\in G$,

The element $e$ is called the identity of the group.

4. Inverses

For every $a\in G$, there exists an element $a^{-1}\in G$ such that

The element $a^{-1}$ is called the inverse of $a$.

We often suppress the operation symbol and write

instead of $a\ast b$. In this notation, the group axioms become

The integers under addition

The most familiar example of a group is

Let $a,b,c\in\mathbb Z$. Then:

and

Therefore $(\mathbb Z,+)$ is a group. Its identity is $0$, and the inverse of $a$ is $-a$.

Because the operation is addition, we usually write the inverse as $-a$ rather than $a^{-1}$. Similarly, repeated addition is written

for $n>0$.

A non-example: the integers under multiplication

Consider $(\mathbb Z,\cdot)$. Multiplication is closed and associative, and $1$ is an identity:

However, most integers do not have multiplicative inverses in $\mathbb Z$. For example, an inverse of $2$ would have to satisfy

so $x=\frac12$, but $\frac12\notin\mathbb Z$. Therefore

is not a group.

If we instead use the nonzero rational numbers,

then $(\mathbb Q^\times,\cdot)$ is a group because every $q\neq0$ has the inverse

We exclude $0$ because no number $x$ can satisfy $0x=1$.

Abelian and non-abelian groups

A group $(G,\ast)$ is abelian if

for all $a,b\in G$.

The groups $(\mathbb Z,+)$ and $(\mathbb Q^\times,\cdot)$ are abelian. Not every group is.

For example, rotations in three-dimensional space generally do not commute. Rotating an object $90^\circ$ around the $x$-axis and then $90^\circ$ around the $y$-axis usually produces a different result from performing the rotations in the opposite order:

The order of operations matters in a non-abelian group.

Basic consequences of the axioms

The group axioms are short, but they imply many useful facts.

The identity is unique

Suppose $e$ and $f$ are both identity elements. Since $e$ is an identity,

Since $f$ is an identity,

Therefore

A group cannot have two different identity elements.

Every inverse is unique

Suppose $b$ and $c$ are both inverses of $a$. Then

and

Using associativity,

Thus the inverse of $a$ is unique.

The inverse of a product

For $a,b\in G$,

The order reverses because

and similarly,

This is analogous to putting on socks and then shoes: to undo the process, we remove the shoes first and the socks second.

Cancellation

If

then multiplying both sides on the left by $a^{-1}$ gives

Associativity now gives

so

and therefore

This is the left cancellation law. Similarly,

Finite groups and Cayley tables

A group is finite if it contains finitely many elements. The number of elements in $G$ is called the order of $G$ and is denoted by

For a finite group, its operation can be displayed in a Cayley table.

Consider the integers modulo $3$ under addition:

Addition is performed modulo $3$, so

and

Its Cayley table is

Every row and every column contains each group element exactly once. This follows from cancellation. For example, if two entries in the row corresponding to $a$ were equal, then

would imply $b=c$.

Subgroups

A subset $H\subseteq G$ is a subgroup of $G$ if $H$ is itself a group under the operation inherited from $G$. We write

Every group has at least two obvious subgroups:

and

The first is called the trivial subgroup.

For example, the even integers form a subgroup of $(\mathbb Z,+)$:

Indeed, if $2m,2n\in2\mathbb Z$, then

and

The identity $0=2\cdot0$ also belongs to $2\mathbb Z$.

The subgroup test

A nonempty subset $H\subseteq G$ is a subgroup if, for every $a,b\in H$,

This single condition contains everything we need.

Taking $a=b$ gives

Now that $e\in H$, taking $a=e$ gives

Finally, since $b^{-1}\in H$, applying the condition to $a$ and $b^{-1}$ gives

Thus $H$ contains the identity, inverses, and products.

Cyclic groups

Let $a\in G$. The subgroup generated by $a$ is

Here,

If

then $G$ is called a cyclic group, and $a$ is called a generator of $G$.

The additive group of integers is cyclic:

In additive notation,

The group $\mathbb Z_6$ is also cyclic:

Starting from $1$ and repeatedly adding $1$ modulo $6$ gives

The element $5$ is another generator because

However,

is only a proper subgroup of $\mathbb Z_6$.

Every cyclic group is abelian. If $x=a^m$ and $y=a^n$, then

The order of an element

The order of an element $a\in G$, written $\operatorname{ord}(a)$, is the smallest positive integer $n$ such that

If no such positive integer exists, then $a$ has infinite order.

In $\mathbb Z_6$, using additive notation, the order of $2$ is $3$ because

while neither $2$ nor $2+2$ is congruent to $0$ modulo $6$.

Thus

More generally, in the additive group $\mathbb Z_n$,

For example,

Indeed,

Permutation groups

A permutation of a set is a bijection from the set to itself. The set of all permutations of

forms the symmetric group $S_n$ under composition.

The group $S_3$ has

elements.

One permutation is

This means

In cycle notation,

Applying $\sigma$ three times returns every element to its starting position:

Therefore

Permutation composition is generally not commutative. Let

Using the convention that the rightmost permutation acts first,

whereas

Hence

so $S_3$ is non-abelian.

Symmetry groups

Group theory gives a precise language for symmetry.

Consider an equilateral triangle. The transformations that leave it looking unchanged are:

• the identity transformation,
• rotations by $120^\circ$ and $240^\circ$,
• and three reflections.

These six symmetries form the dihedral group

Let $r$ denote rotation by $120^\circ$ and let $s$ denote a reflection. Then

and the interaction between rotation and reflection is described by

Equivalently,

The group can be written as

Since

the group $D_3$ is non-abelian. In fact,

The symbol $\cong$ means that the two groups have the same abstract group structure even though their elements are described differently.

Homomorphisms

A group homomorphism is a function

that preserves the group operation:

for all $a,b\in G$.

If the groups use additive notation, this becomes

For example, define

by

Then

so $\varphi$ is a homomorphism.

Every homomorphism sends the identity to the identity:

To see this, observe that

Cancellation in $H$ gives

Homomorphisms also preserve inverses:

Indeed,

Kernel and image

For a homomorphism $\varphi\colon G\to H$, its kernel is

and its image is

For the reduction map

we have

because

Also,

because every congruence class $[r]_n$ is the image of the integer $r$.

The kernel measures which elements become indistinguishable under the homomorphism. A homomorphism is injective exactly when

Cosets

Let $H\leq G$ and $g\in G$. The left coset of $H$ determined by $g$ is

The corresponding right coset is

In an additive group, we write

Consider the subgroup

of $\mathbb Z$. Its cosets are

and

Every integer belongs to exactly one of these three cosets. They are precisely the three congruence classes modulo $3$.

Every coset $gH$ has the same number of elements as $H$ because the function

is a bijection.

Lagrange’s theorem

One of the first major results in finite group theory is Lagrange’s theorem:

If $G$ is a finite group and $H\leq G$, then

$$|H|\mid|G|.$$

In other words, the order of every subgroup divides the order of the group.

The reason is that the left cosets of $H$ partition $G$, and every coset contains exactly $|H|$ elements. If there are $[G:H]$ distinct cosets, then

The number $[G:H]$ is called the index of $H$ in $G$.

As an immediate consequence, the order of every element divides the order of the group:

For example, a group of order $12$ cannot contain an element of order $5$, because

If $|G|=p$ for a prime number $p$, then every non-identity element has order $p$. Therefore every group of prime order is cyclic:

Normal subgroups and quotient groups

A subgroup $N\leq G$ is normal if

for every $g\in G$. We write

An equivalent condition is

for every $g\in G$.

Every subgroup of an abelian group is normal because

for all $g\in G$ and $n\in N$.

When $N\trianglelefteq G$, the set of cosets

becomes a group under

For example,

is the quotient of $\mathbb Z$ by the subgroup $n\mathbb Z$. Its elements are

This connects modular arithmetic directly with group theory.

A compact example

Consider the additive group

Let

Repeatedly adding $4$ gives

so

and

The distinct cosets of $H$ are

Therefore

and Lagrange’s theorem appears concretely as

Because $G$ is abelian, $H$ is normal, and the quotient group has order

In fact,

Exercises

Exercise 1

Show that

is a subgroup of $(\mathbb Z,+)$.

Exercise 2

Find every generator of $\mathbb Z_{10}$.

Exercise 3

Calculate the order of every element of $\mathbb Z_8$.

Exercise 4

Let

Find $\ker\varphi$ and $\operatorname{im}\varphi$.

Exercise 5

Let $G$ be a finite group with $|G|=15$. What element orders are permitted by Lagrange’s theorem?

Answers

For Exercise 1, the set is nonempty, and for $3m,3n\in3\mathbb Z$,

Therefore the subgroup test gives

For Exercise 2, an element $k\in\mathbb Z_{10}$ generates the group exactly when

Hence the generators are

For Exercise 3, use

Thus

For Exercise 4,

and

For Exercise 5, the order of an element must divide $15$. Therefore the only possibilities are

Lagrange’s theorem permits these orders, although it does not by itself guarantee that elements of every permitted order exist.

Where to go next

The basic objects of group theory now fit into one picture:

At the same time, homomorphisms compare different groups:

The next natural topics are:

• group actions,
• the isomorphism theorems,
• direct products,
• the classification of finite abelian groups,
• Sylow’s theorems,
• and presentations by generators and relations.

The main habit to develop is to move constantly between abstract axioms and concrete examples. Groups become intuitive when we see the same structure in integers, permutations, matrices, and geometric symmetries.