Groups: Definitions and First Properties

Starting from the axiomatic definition of a group, we establish the uniqueness of the identity and inverses, cancellation laws, and the laws of exponents. We then develop the theory of subgroups, cyclic groups, and orders of elements, laying the rigorous foundations for everything that follows.

Folio Official

March 1, 2026

1 The Group Axioms

Definition 1 (Group).

A group is a pair

(G, \cdot)

consisting of a set

G

and a binary operation

\cdot : G \times G \to G

satisfying the following three axioms:

Associativity. For all $a, b, c \in G$ , $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ .
Identity element. There exists an element $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$ .
Inverses. For every $a \in G$ , there exists an element $b \in G$ such that $a \cdot b = b \cdot a = e$ .

Throughout, we write the group operation by juxtaposition ( $ab$ instead of $a \cdot b$ ), denote the identity element by $e$ , and denote the inverse of $a$ by $a^{- 1}$ .

Definition 2 (Abelian group).

A group

(G, \cdot)

is called abelian (or commutative) if

ab = ba

for all

a, b \in G

2 Fundamental Examples

Let us ground the abstract definition with some concrete examples.

Example 3 (The additive group of the integers).

(Z, +)

is a group. Associativity is clear, the identity element is

0

, and the inverse of

a

- a

. Since

a + b = b + a

, it is abelian.

Example 4 (The general linear group).

G L_{n} (R) = {A \in M_{n} (R) ∣ det A \neq = 0}

is a group under matrix multiplication. The identity element is the identity matrix

I_{n}

, and the inverse is the matrix inverse

A^{- 1}

. For

n \geq 2

, there exist matrices

A, B

with

A B \neq = B A

, so this group is non-abelian.

Example 5 (The symmetric group).

Let

S_{n}

denote the set of all bijections from

{1, 2, \dots, n}

to itself. Under composition of maps,

S_{n}

forms a group called the symmetric group on $n$ letters. Its order is

∣ S_{n} ∣ = n!

, and it is non-abelian for

n \geq 3

Example 6 (Cyclic groups of integers modulo

n

For a positive integer

n

, define

Z / n Z = {\overset{ˉ}{0}, \overset{ˉ}{1}, \dots, \overline{n - 1}}

with the operation

\overset{a}{ˉ} + \overset{ˉ}{b} = \overline{a + b}

. This is an abelian group.

Example 7 (The trivial group).

The set

{e}

with

e \cdot e = e

forms a group, called the trivial group.

3 Uniqueness of the Identity and Inverses

The group axioms only assert the existence of an identity element and of inverses. However, they are in fact unique.

Theorem 8 (Uniqueness of the identity).

The identity element of a group

G

is unique.

Proof.

Suppose

e

and

e^{'}

are both identity elements. Since

e

is an identity,

e \cdot e^{'} = e^{'}

. Since

e^{'}

is an identity,

e \cdot e^{'} = e

. Therefore

e = e^{'}

. □

Theorem 9 (Uniqueness of inverses).

For each element

a

of a group

G

, the inverse

a^{- 1}

is unique.

Proof.

Suppose

b

and

c

are both inverses of

a

, so that

ab = ba = e

and

a c = c a = e

. Then

b = b e = b (a c) = (ba) c = ec = c .

□

Theorem 10 (Inverse of the inverse).

For any element

a

of a group

G

(a^{- 1})^{- 1} = a

Proof.

The inverse of

a^{- 1}

is the element

x

satisfying

a^{- 1} x = x a^{- 1} = e

. Since

a^{- 1} a = a a^{- 1} = e

, we have

x = a

. □

Theorem 11 (Inverse of a product).

For any elements

a, b

of a group

G

(ab)^{- 1} = b^{- 1} a^{- 1}

Proof.

We verify directly:

(ab) (b^{- 1} a^{- 1}) = a (b b^{- 1}) a^{- 1} = a e a^{- 1} = a a^{- 1} = e

. Similarly,

(b^{- 1} a^{- 1}) (ab) = e

. □

4 Cancellation Laws

Theorem 12 (Cancellation laws).

In a group

G

, the following hold:

Left cancellation: $ab = a c \Rightarrow b = c$ .
Right cancellation: $ba = c a \Rightarrow b = c$ .

Proof.

(1) If

ab = a c

, multiply both sides on the left by

a^{- 1}

a^{- 1} (ab) = a^{- 1} (a c) ⟹ (a^{- 1} a) b = (a^{- 1} a) c ⟹ e b = ec ⟹ b = c .

(2) Similarly, multiply on the right by

a^{- 1}

. □

Theorem 13 (Solubility of equations).

In a group

G

, for any

a, b \in G

, the equations

a x = b

and

y a = b

each have a unique solution.

Proof.

The equation

a x = b

has the solution

x = a^{- 1} b

. Indeed,

a (a^{- 1} b) = (a a^{- 1}) b = e b = b

. Uniqueness follows from the cancellation law. Similarly,

y = b a^{- 1}

is the unique solution to

y a = b

. □

Remark 14.

This theorem captures an essential characterization of groups: a semigroup (a set with an associative binary operation) in which the equations

a x = b

and

y a = b

are always solvable is, in fact, a group. Some authors adopt this as an alternative axiom system.

5 Powers and Order of an Element

Definition 15 (Powers).

For an element

a

of a group

G

and an integer

n

, we define the power

a^{n}

as follows:

$a^{0} = e$ .
For $n > 0$ , $a^{n} = n factors a \cdot a \dots a$ .
For $n < 0$ , $a^{n} = (a^{- 1})^{∣ n ∣}$ .

Theorem 16 (Laws of exponents).

For an element

a

of a group

G

and integers

m, n

, the following hold:

$a^{m + n} = a^{m} a^{n}$ .
$(a^{m})^{n} = a^{mn}$ .

Proof.

(1) When

m, n \geq 0

, this follows by induction on

n

. When

m < 0

and

n \geq 0

with

∣ m ∣ \leq n

, we have

a^{m} a^{n} = (a^{- 1})^{∣ m ∣} a^{n} = a^{n - ∣ m ∣} = a^{m + n}

. The remaining cases are handled by similar case analysis. (2) Follows by the same method. □

Definition 17 (Order of an element).

For an element

a

of a group

G

, if there exists a smallest positive integer

n

such that

a^{n} = e

, then

n

is called the order of

a

, and we write

ord (a) = n

∣ a ∣ = n

. If no such positive integer exists, we say

a

has infinite order and write

ord (a) = \infty

Example 18.

Z /6 Z

, we have

ord (\overset{ˉ}{1}) = 6

ord (\overset{ˉ}{2}) = 3

ord (\overset{ˉ}{3}) = 2

, and

ord (\overset{ˉ}{0}) = 1

Example 19.

(Z, +)

, every nonzero element has infinite order.

Theorem 20 (Order and powers).

Let

ord (a) = n < \infty

. Then

a^{k} = e

if and only if

n ∣ k

Proof.

(\Leftarrow)

: If

k = n q

, then

a^{k} = (a^{n})^{q} = e^{q} = e

(\Rightarrow)

: Write

k = n q + r

with

0 \leq r < n

. From

a^{k} = e

we get

a^{r} = a^{k - n q} = a^{k} (a^{n})^{- q} = e \cdot e = e

. Since

r < n

and

n

is the smallest positive integer with

a^{n} = e

, we must have

r = 0

, i.e.

n ∣ k

. □

6 Subgroups

Definition 21 (Subgroup).

A nonempty subset

H

of a group

G

is called a subgroup of

G

H

is itself a group under the operation of

G

. We write

H \leq G

. If

H \neq = {e}

and

H \neq = G

, we say

H

is a proper subgroup and write

H < G

Checking the subgroup axioms directly from the definition is often cumbersome. The following criterion provides a convenient shortcut.

Theorem 22 (One-step subgroup test).

A nonempty subset

H

of a group

G

is a subgroup if and only if

a b^{- 1} \in H

for all

a, b \in H

Proof.

(\Rightarrow)

: If

H

is a subgroup, then

b \in H

implies

b^{- 1} \in H

, and then

a, b^{- 1} \in H

implies

a b^{- 1} \in H

(\Leftarrow)

: Since

H \neq = \emptyset

, pick any

a \in H

. Taking

b = a

gives

a a^{- 1} = e \in H

. Taking

a = e

gives

e b^{- 1} = b^{- 1} \in H

for any

b \in H

. For

a, b \in H

, since

b^{- 1} \in H

, we have

a (b^{- 1})^{- 1} = ab \in H

(closure). Associativity is inherited from

G

. □

Theorem 23 (Finite subgroup test).

A nonempty finite subset

H

of a group

G

is a subgroup if and only if

H

is closed under the group operation (i.e.

a, b \in H \Rightarrow ab \in H

Proof.

(\Rightarrow)

is clear.

(\Leftarrow)

: Let

a \in H

. The elements

a, a^{2}, a^{3}, \dots

all lie in

H

, and since

H

is finite, we must have

a^{i} = a^{j}

for some

i < j

. By cancellation,

a^{j - i} = e

, so

e \in H

. If

j - i \geq 2

, then

a^{j - i - 1} = a^{- 1} \in H

. If

j - i = 1

, then

a = e

, so

a^{- 1} = e \in H

. □

Example 24 (Examples of subgroups).

For any group $G$ , the subsets ${e}$ and $G$ are subgroups (trivial subgroups).
The subgroups of $(Z, +)$ are precisely the sets $n Z = {nk ∣ k \in Z}$ for $n = 0, 1, 2, \dots$ (proved below).
$S L_{n} (R) = {A \in G L_{n} (R) ∣ det A = 1}$ is a subgroup of $G L_{n} (R)$ (the special linear group).
The set $A_{n}$ of all even permutations is a subgroup of $S_{n}$ (the alternating group), with $∣ A_{n} ∣ = n! /2$ .

Theorem 25 (Intersection of subgroups).

{H_{i}}_{i \in I}

is a family of subgroups of

G

, then

⋂_{i \in I} H_{i}

is also a subgroup of

G

Proof.

Since

e \in H_{i}

for all

i

, we have

⋂ H_{i} \neq = \emptyset

. If

a, b \in ⋂ H_{i}

, then

a, b \in H_{i}

for each

i

, and since each

H_{i}

is a subgroup,

a b^{- 1} \in H_{i}

. Therefore

a b^{- 1} \in ⋂ H_{i}

. □

Remark 26.

The union of subgroups is not, in general, a subgroup. For example,

2 Z \cup 3 Z

is not a subgroup of

Z

, since

2 + 3 = 5 \in / 2 Z \cup 3 Z

7 Generators and Cyclic Groups

Definition 27 (Generated subgroup).

For a subset

S

of a group

G

, the subgroup generated by $S$ is defined as the intersection of all subgroups of

G

containing

S

⟨ S ⟩ = S \subseteq H \leq G ⋂ H

Theorem 28.

⟨ S ⟩

equals the set of all finite products of elements of

S

and their inverses:

⟨ S ⟩ = {a_{1}^{ε_{1}} a_{2}^{ε_{2}} \dots a_{k}^{ε_{k}} ∣ k \geq 0, a_{i} \in S, ε_{i} = \pm 1}

where the empty product (when

k = 0

) is defined to be

e

Proof.

Let

T

denote the right-hand side. We must verify three things: that

T

is a subgroup, that

S \subseteq T

, and that

T

is contained in every subgroup that contains

S

T

is closed under the group operation (concatenate two products). We have

e \in T

(take

k = 0

). The inverse of

a_{1}^{ε_{1}} \dots a_{k}^{ε_{k}}

a_{k}^{- ε_{k}} \dots a_{1}^{- ε_{1}} \in T

. Thus

T

is a subgroup.

For

a \in S

, taking

k = 1

and

ε_{1} = 1

gives

a \in T

, so

S \subseteq T

S \subseteq H \leq G

, then

H

is closed under products and inverses, so

T \subseteq H

. Therefore

T = ⟨ S ⟩

. □

Definition 29 (Cyclic group).

A group

G

is cyclic if

G = ⟨ a ⟩

for some element

a \in G

. The element

a

is called a generator of

G

The cyclic group $⟨ a ⟩$ is precisely the set ${a^{n} ∣ n \in Z}$ of all powers of $a$ .

Theorem 30 (Structure of cyclic groups).

If $ord (a) = \infty$ , then $⟨ a ⟩ ≅ Z$ (as additive groups).
If $ord (a) = n < \infty$ , then $⟨ a ⟩ = {e, a, a^{2}, \dots, a^{n - 1}}$ , $∣ ⟨ a ⟩ ∣ = n$ , and $⟨ a ⟩ ≅ Z / n Z$ .

Proof.

(1) Define

φ : Z \to ⟨ a ⟩

φ (k) = a^{k}

. Then

φ (m + n) = a^{m + n} = a^{m} a^{n} = φ (m) φ (n)

, so

φ

is a homomorphism. Since

ord (a) = \infty

, if

a^{m} = a^{n}

then

a^{m - n} = e

, which forces

m = n

. So

φ

is injective. Surjectivity is immediate from the definition. Hence

φ

is an isomorphism.

(2) We first show

e, a, a^{2}, \dots, a^{n - 1}

are all distinct. If

a^{i} = a^{j}

with

0 \leq i < j \leq n - 1

, then

a^{j - i} = e

with

0 < j - i < n

, contradicting the minimality of

n

. So

∣ ⟨ a ⟩ ∣ = n

. The isomorphism is given by

φ : Z / n Z \to ⟨ a ⟩

φ (\overset{ˉ}{k}) = a^{k}

; well-definedness follows from the theorem relating order and powers. □

Theorem 31 (Subgroups of cyclic groups).

Every subgroup of a cyclic group is cyclic.

Proof.

Let

G = ⟨ a ⟩

and let

H \leq G

. If

H = {e}

, then

H = ⟨ e ⟩

is cyclic. Suppose

H \neq = {e}

. Let

d

be the smallest positive integer such that

a^{d} \in H

. (Such a

d

exists because

H

contains some

a^{k}

with

k \neq = 0

, and

a^{k} \in H

implies

a^{- k} \in H

, so we can always find a positive power.)

a^{m} \in H

, write

m = d q + r

with

0 \leq r < d

. Then

a^{r} = a^{m} (a^{d})^{- q} \in H

. By the minimality of

d

, we must have

r = 0

, i.e.

d ∣ m

. Therefore

H = ⟨ a^{d} ⟩

. □

Theorem 32 (Number of generators of a cyclic group).

For a cyclic group

G = ⟨ a ⟩

of order

n

, the element

a^{k}

is a generator of

G

if and only if

g cd (k, n) = 1

. Consequently,

G

has exactly

φ (n)

generators, where

φ

is Euler's totient function.

Proof.

We show that

ord (a^{k}) = n / g cd (k, n)

. Let

d = g cd (k, n)

. We have

(a^{k})^{n / d} = a^{kn / d} = (a^{n})^{k / d} = e

, so

ord (a^{k}) ∣ n / d

. Conversely, if

(a^{k})^{m} = e

, then

n ∣ km

, i.e.

(n / d) ∣ (k / d) m

. Since

g cd (n / d, k / d) = 1

, it follows that

(n / d) ∣ m

. Therefore

ord (a^{k}) = n / d

. Now

ord (a^{k}) = n

if and only if

d = 1

, i.e.

g cd (k, n) = 1

. □

8 Order of a Group

Definition 33 (Order of a group).

The order of a group

G

, denoted

∣ G ∣

, is the cardinality of the underlying set. If

∣ G ∣ < \infty

, we call

G

a finite group; otherwise,

G

is an infinite group.

Example 34.

∣ S_{n} ∣ = n!

∣ Z / n Z ∣ = n

∣ D_{n} ∣ = 2 n

(the dihedral group of a regular

n

-gon), and

∣ (Z, +) ∣ = \infty

Theorem 35 (Order of an element divides the order of the group).

For an element

a

of a finite group

G

ord (a)

divides

∣ G ∣

. In particular,

a^{∣ G ∣} = e

Proof.

The cyclic subgroup

⟨ a ⟩

is a subgroup of

G

with

∣ ⟨ a ⟩ ∣ = ord (a)

. By Lagrange's theorem (proved in the next chapter),

∣ ⟨ a ⟩ ∣

divides

∣ G ∣

. Writing

∣ G ∣ = ord (a) \cdot m

, we get

a^{∣ G ∣} = (a^{ord (a)})^{m} = e^{m} = e

. □

9 Fundamentals of Group Homomorphisms

We will study homomorphisms in depth later, but it is convenient to state the definition and basic properties now.

Definition 36 (Group homomorphism).

A map

φ : G \to G^{'}

between groups is called a group homomorphism if

φ (ab) = φ (a) φ (b) for all a, b \in G .

Theorem 37 (Basic properties of homomorphisms).

Let

φ : G \to G^{'}

be a group homomorphism. Then:

$φ (e_{G}) = e_{G^{'}}$ .
$φ (a^{- 1}) = φ (a)^{- 1}$ .
$φ (a^{n}) = φ (a)^{n}$ for all $n \in Z$ .

Proof.

(1) We have

φ (e_{G}) = φ (e_{G} \cdot e_{G}) = φ (e_{G}) φ (e_{G})

. Multiplying both sides by

φ (e_{G})^{- 1}

yields

e_{G^{'}} = φ (e_{G})

(2)

φ (a) φ (a^{- 1}) = φ (a a^{- 1}) = φ (e_{G}) = e_{G^{'}}

, so

φ (a^{- 1}) = φ (a)^{- 1}

(3) For

n \geq 0

, this follows by induction. For

n < 0

, use part (2). □

Definition 38 (Kernel and image).

For a group homomorphism

φ : G \to G^{'}

The kernel of $φ$ is $ker φ = {a \in G ∣ φ (a) = e_{G^{'}}}$ .
The image of $φ$ is $Im φ = {φ (a) ∣ a \in G}$ .

One can verify that

ker φ \leq G

and

Im φ \leq G^{'}

Theorem 39 (Injectivity and the kernel).

A group homomorphism

φ : G \to G^{'}

is injective if and only if

ker φ = {e_{G}}

Proof.

(\Rightarrow)

: If

φ (a) = e_{G^{'}} = φ (e_{G})

, then injectivity gives

a = e_{G}

(\Leftarrow)

: If

φ (a) = φ (b)

, then

φ (a b^{- 1}) = φ (a) φ (b)^{- 1} = e_{G^{'}}

, so

a b^{- 1} \in ker φ = {e_{G}}

, whence

a = b

. □

10 Summary and Next Steps

In this chapter we have systematically developed the foundations of group theory:

The group axioms and fundamental examples.
Uniqueness of the identity and inverses, cancellation laws, and solubility of equations.
Laws of exponents and the order of an element.
The definition of subgroups and criteria for recognizing them.
The structure of cyclic groups and their subgroups.
Group homomorphisms: definition, kernel, image, and the injectivity criterion.

In the next chapter, we consider how a subgroup "partitions" a group. This leads to the concept of cosets and to Lagrange's theorem— the most fundamental theorem in the theory of finite groups.

Group Theory Algebra Textbook Group Definition Subgroups Cyclic Groups

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Groups: Definitions and First Properties

Folio Official

March 1, 2026

1 The Group Axioms

Definition 1 (Group).

A group is a pair

(G, \cdot)

consisting of a set

G

and a binary operation

\cdot : G \times G \to G

satisfying the following three axioms:

Associativity. For all $a, b, c \in G$ , $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ .
Identity element. There exists an element $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$ .
Inverses. For every $a \in G$ , there exists an element $b \in G$ such that $a \cdot b = b \cdot a = e$ .

Throughout, we write the group operation by juxtaposition ( $ab$ instead of $a \cdot b$ ), denote the identity element by $e$ , and denote the inverse of $a$ by $a^{- 1}$ .

Definition 2 (Abelian group).

A group

(G, \cdot)

is called abelian (or commutative) if

ab = ba

for all

a, b \in G

2 Fundamental Examples

Let us ground the abstract definition with some concrete examples.

Example 3 (The additive group of the integers).

(Z, +)

is a group. Associativity is clear, the identity element is

0

, and the inverse of

a

- a

. Since

a + b = b + a

, it is abelian.

Example 4 (The general linear group).

G L_{n} (R) = {A \in M_{n} (R) ∣ det A \neq = 0}

is a group under matrix multiplication. The identity element is the identity matrix

I_{n}

, and the inverse is the matrix inverse

A^{- 1}

. For

n \geq 2

, there exist matrices

A, B

with

A B \neq = B A

, so this group is non-abelian.

Example 5 (The symmetric group).

Let

S_{n}

denote the set of all bijections from

{1, 2, \dots, n}

to itself. Under composition of maps,

S_{n}

forms a group called the symmetric group on $n$ letters. Its order is

∣ S_{n} ∣ = n!

, and it is non-abelian for

n \geq 3

Example 6 (Cyclic groups of integers modulo

n

For a positive integer

n

, define

Z / n Z = {\overset{ˉ}{0}, \overset{ˉ}{1}, \dots, \overline{n - 1}}

with the operation

\overset{a}{ˉ} + \overset{ˉ}{b} = \overline{a + b}

. This is an abelian group.

Example 7 (The trivial group).

The set

{e}

with

e \cdot e = e

forms a group, called the trivial group.

3 Uniqueness of the Identity and Inverses

The group axioms only assert the existence of an identity element and of inverses. However, they are in fact unique.

Theorem 8 (Uniqueness of the identity).

The identity element of a group

G

is unique.

Proof.

Suppose

e

and

e^{'}

are both identity elements. Since

e

is an identity,

e \cdot e^{'} = e^{'}

. Since

e^{'}

is an identity,

e \cdot e^{'} = e

. Therefore

e = e^{'}

. □

Theorem 9 (Uniqueness of inverses).

For each element

a

of a group

G

, the inverse

a^{- 1}

is unique.

Proof.

Suppose

b

and

c

are both inverses of

a

, so that

ab = ba = e

and

a c = c a = e

. Then

b = b e = b (a c) = (ba) c = ec = c .

□

Theorem 10 (Inverse of the inverse).

For any element

a

of a group

G

(a^{- 1})^{- 1} = a

Proof.

The inverse of

a^{- 1}

is the element

x

satisfying

a^{- 1} x = x a^{- 1} = e

. Since

a^{- 1} a = a a^{- 1} = e

, we have

x = a

. □

Theorem 11 (Inverse of a product).

For any elements

a, b

of a group

G

(ab)^{- 1} = b^{- 1} a^{- 1}

Proof.

We verify directly:

(ab) (b^{- 1} a^{- 1}) = a (b b^{- 1}) a^{- 1} = a e a^{- 1} = a a^{- 1} = e

. Similarly,

(b^{- 1} a^{- 1}) (ab) = e

. □

4 Cancellation Laws

Theorem 12 (Cancellation laws).

In a group

G

, the following hold:

Left cancellation: $ab = a c \Rightarrow b = c$ .
Right cancellation: $ba = c a \Rightarrow b = c$ .

Proof.

(1) If

ab = a c

, multiply both sides on the left by

a^{- 1}

a^{- 1} (ab) = a^{- 1} (a c) ⟹ (a^{- 1} a) b = (a^{- 1} a) c ⟹ e b = ec ⟹ b = c .

(2) Similarly, multiply on the right by

a^{- 1}

. □

Theorem 13 (Solubility of equations).

In a group

G

, for any

a, b \in G

, the equations

a x = b

and

y a = b

each have a unique solution.

Proof.

The equation

a x = b

has the solution

x = a^{- 1} b

. Indeed,

a (a^{- 1} b) = (a a^{- 1}) b = e b = b

. Uniqueness follows from the cancellation law. Similarly,

y = b a^{- 1}

is the unique solution to

y a = b

. □

Remark 14.

This theorem captures an essential characterization of groups: a semigroup (a set with an associative binary operation) in which the equations

a x = b

and

y a = b

are always solvable is, in fact, a group. Some authors adopt this as an alternative axiom system.

5 Powers and Order of an Element

Definition 15 (Powers).

For an element

a

of a group

G

and an integer

n

, we define the power

a^{n}

as follows:

$a^{0} = e$ .
For $n > 0$ , $a^{n} = n factors a \cdot a \dots a$ .
For $n < 0$ , $a^{n} = (a^{- 1})^{∣ n ∣}$ .

Theorem 16 (Laws of exponents).

For an element

a

of a group

G

and integers

m, n

, the following hold:

$a^{m + n} = a^{m} a^{n}$ .
$(a^{m})^{n} = a^{mn}$ .

Proof.

(1) When

m, n \geq 0

, this follows by induction on

n

. When

m < 0

and

n \geq 0

with

∣ m ∣ \leq n

, we have

a^{m} a^{n} = (a^{- 1})^{∣ m ∣} a^{n} = a^{n - ∣ m ∣} = a^{m + n}

. The remaining cases are handled by similar case analysis. (2) Follows by the same method. □

Definition 17 (Order of an element).

For an element

a

of a group

G

, if there exists a smallest positive integer

n

such that

a^{n} = e

, then

n

is called the order of

a

, and we write

ord (a) = n

∣ a ∣ = n

. If no such positive integer exists, we say

a

has infinite order and write

ord (a) = \infty

Example 18.

Z /6 Z

, we have

ord (\overset{ˉ}{1}) = 6

ord (\overset{ˉ}{2}) = 3

ord (\overset{ˉ}{3}) = 2

, and

ord (\overset{ˉ}{0}) = 1

Example 19.

(Z, +)

, every nonzero element has infinite order.

Theorem 20 (Order and powers).

Let

ord (a) = n < \infty

. Then

a^{k} = e

if and only if

n ∣ k

Proof.

(\Leftarrow)

: If

k = n q

, then

a^{k} = (a^{n})^{q} = e^{q} = e

(\Rightarrow)

: Write

k = n q + r

with

0 \leq r < n

. From

a^{k} = e

we get

a^{r} = a^{k - n q} = a^{k} (a^{n})^{- q} = e \cdot e = e

. Since

r < n

and

n

is the smallest positive integer with

a^{n} = e

, we must have

r = 0

, i.e.

n ∣ k

. □

6 Subgroups

Definition 21 (Subgroup).

A nonempty subset

H

of a group

G

is called a subgroup of

G

H

is itself a group under the operation of

G

. We write

H \leq G

. If

H \neq = {e}

and

H \neq = G

, we say

H

is a proper subgroup and write

H < G

Checking the subgroup axioms directly from the definition is often cumbersome. The following criterion provides a convenient shortcut.

Theorem 22 (One-step subgroup test).

A nonempty subset

H

of a group

G

is a subgroup if and only if

a b^{- 1} \in H

for all

a, b \in H

Proof.

(\Rightarrow)

: If

H

is a subgroup, then

b \in H

implies

b^{- 1} \in H

, and then

a, b^{- 1} \in H

implies

a b^{- 1} \in H

(\Leftarrow)

: Since

H \neq = \emptyset

, pick any

a \in H

. Taking

b = a

gives

a a^{- 1} = e \in H

. Taking

a = e

gives

e b^{- 1} = b^{- 1} \in H

for any

b \in H

. For

a, b \in H

, since

b^{- 1} \in H

, we have

a (b^{- 1})^{- 1} = ab \in H

(closure). Associativity is inherited from

G

. □

Theorem 23 (Finite subgroup test).

A nonempty finite subset

H

of a group

G

is a subgroup if and only if

H

is closed under the group operation (i.e.

a, b \in H \Rightarrow ab \in H

Proof.

(\Rightarrow)

is clear.

(\Leftarrow)

: Let

a \in H

. The elements

a, a^{2}, a^{3}, \dots

all lie in

H

, and since

H

is finite, we must have

a^{i} = a^{j}

for some

i < j

. By cancellation,

a^{j - i} = e

, so

e \in H

. If

j - i \geq 2

, then

a^{j - i - 1} = a^{- 1} \in H

. If

j - i = 1

, then

a = e

, so

a^{- 1} = e \in H

. □

Example 24 (Examples of subgroups).

For any group $G$ , the subsets ${e}$ and $G$ are subgroups (trivial subgroups).
The subgroups of $(Z, +)$ are precisely the sets $n Z = {nk ∣ k \in Z}$ for $n = 0, 1, 2, \dots$ (proved below).
$S L_{n} (R) = {A \in G L_{n} (R) ∣ det A = 1}$ is a subgroup of $G L_{n} (R)$ (the special linear group).
The set $A_{n}$ of all even permutations is a subgroup of $S_{n}$ (the alternating group), with $∣ A_{n} ∣ = n! /2$ .

Theorem 25 (Intersection of subgroups).

{H_{i}}_{i \in I}

is a family of subgroups of

G

, then

⋂_{i \in I} H_{i}

is also a subgroup of

G

Proof.

Since

e \in H_{i}

for all

i

, we have

⋂ H_{i} \neq = \emptyset

. If

a, b \in ⋂ H_{i}

, then

a, b \in H_{i}

for each

i

, and since each

H_{i}

is a subgroup,

a b^{- 1} \in H_{i}

. Therefore

a b^{- 1} \in ⋂ H_{i}

. □

Remark 26.

The union of subgroups is not, in general, a subgroup. For example,

2 Z \cup 3 Z

is not a subgroup of

Z

, since

2 + 3 = 5 \in / 2 Z \cup 3 Z

7 Generators and Cyclic Groups

Definition 27 (Generated subgroup).

For a subset

S

of a group

G

, the subgroup generated by $S$ is defined as the intersection of all subgroups of

G

containing

S

⟨ S ⟩ = S \subseteq H \leq G ⋂ H

Theorem 28.

⟨ S ⟩

equals the set of all finite products of elements of

S

and their inverses:

⟨ S ⟩ = {a_{1}^{ε_{1}} a_{2}^{ε_{2}} \dots a_{k}^{ε_{k}} ∣ k \geq 0, a_{i} \in S, ε_{i} = \pm 1}

where the empty product (when

k = 0

) is defined to be

e

Proof.

Let

T

denote the right-hand side. We must verify three things: that

T

is a subgroup, that

S \subseteq T

, and that

T

is contained in every subgroup that contains

S

T

is closed under the group operation (concatenate two products). We have

e \in T

(take

k = 0

). The inverse of

a_{1}^{ε_{1}} \dots a_{k}^{ε_{k}}

a_{k}^{- ε_{k}} \dots a_{1}^{- ε_{1}} \in T

. Thus

T

is a subgroup.

For

a \in S

, taking

k = 1

and

ε_{1} = 1

gives

a \in T

, so

S \subseteq T

S \subseteq H \leq G

, then

H

is closed under products and inverses, so

T \subseteq H

. Therefore

T = ⟨ S ⟩

. □

Definition 29 (Cyclic group).

A group

G

is cyclic if

G = ⟨ a ⟩

for some element

a \in G

. The element

a

is called a generator of

G

The cyclic group $⟨ a ⟩$ is precisely the set ${a^{n} ∣ n \in Z}$ of all powers of $a$ .

Theorem 30 (Structure of cyclic groups).

If $ord (a) = \infty$ , then $⟨ a ⟩ ≅ Z$ (as additive groups).
If $ord (a) = n < \infty$ , then $⟨ a ⟩ = {e, a, a^{2}, \dots, a^{n - 1}}$ , $∣ ⟨ a ⟩ ∣ = n$ , and $⟨ a ⟩ ≅ Z / n Z$ .

Proof.

(1) Define

φ : Z \to ⟨ a ⟩

φ (k) = a^{k}

. Then

φ (m + n) = a^{m + n} = a^{m} a^{n} = φ (m) φ (n)

, so

φ

is a homomorphism. Since

ord (a) = \infty

, if

a^{m} = a^{n}

then

a^{m - n} = e

, which forces

m = n

. So

φ

is injective. Surjectivity is immediate from the definition. Hence

φ

is an isomorphism.

(2) We first show

e, a, a^{2}, \dots, a^{n - 1}

are all distinct. If

a^{i} = a^{j}

with

0 \leq i < j \leq n - 1

, then

a^{j - i} = e

with

0 < j - i < n

, contradicting the minimality of

n

. So

∣ ⟨ a ⟩ ∣ = n

. The isomorphism is given by

φ : Z / n Z \to ⟨ a ⟩

φ (\overset{ˉ}{k}) = a^{k}

; well-definedness follows from the theorem relating order and powers. □

Theorem 31 (Subgroups of cyclic groups).

Every subgroup of a cyclic group is cyclic.

Proof.

Let

G = ⟨ a ⟩

and let

H \leq G

. If

H = {e}

, then

H = ⟨ e ⟩

is cyclic. Suppose

H \neq = {e}

. Let

d

be the smallest positive integer such that

a^{d} \in H

. (Such a

d

exists because

H

contains some

a^{k}

with

k \neq = 0

, and

a^{k} \in H

implies

a^{- k} \in H

, so we can always find a positive power.)

a^{m} \in H

, write

m = d q + r

with

0 \leq r < d

. Then

a^{r} = a^{m} (a^{d})^{- q} \in H

. By the minimality of

d

, we must have

r = 0

, i.e.

d ∣ m

. Therefore

H = ⟨ a^{d} ⟩

. □

Theorem 32 (Number of generators of a cyclic group).

For a cyclic group

G = ⟨ a ⟩

of order

n

, the element

a^{k}

is a generator of

G

if and only if

g cd (k, n) = 1

. Consequently,

G

has exactly

φ (n)

generators, where

φ

is Euler's totient function.

Proof.

We show that

ord (a^{k}) = n / g cd (k, n)

. Let

d = g cd (k, n)

. We have

(a^{k})^{n / d} = a^{kn / d} = (a^{n})^{k / d} = e

, so

ord (a^{k}) ∣ n / d

. Conversely, if

(a^{k})^{m} = e

, then

n ∣ km

, i.e.

(n / d) ∣ (k / d) m

. Since

g cd (n / d, k / d) = 1

, it follows that

(n / d) ∣ m

. Therefore

ord (a^{k}) = n / d

. Now

ord (a^{k}) = n

if and only if

d = 1

, i.e.

g cd (k, n) = 1

. □

8 Order of a Group

Definition 33 (Order of a group).

The order of a group

G

, denoted

∣ G ∣

, is the cardinality of the underlying set. If

∣ G ∣ < \infty

, we call

G

a finite group; otherwise,

G

is an infinite group.

Example 34.

∣ S_{n} ∣ = n!

∣ Z / n Z ∣ = n

∣ D_{n} ∣ = 2 n

(the dihedral group of a regular

n

-gon), and

∣ (Z, +) ∣ = \infty

Theorem 35 (Order of an element divides the order of the group).

For an element

a

of a finite group

G

ord (a)

divides

∣ G ∣

. In particular,

a^{∣ G ∣} = e

Proof.

The cyclic subgroup

⟨ a ⟩

is a subgroup of

G

with

∣ ⟨ a ⟩ ∣ = ord (a)

. By Lagrange's theorem (proved in the next chapter),

∣ ⟨ a ⟩ ∣

divides

∣ G ∣

. Writing

∣ G ∣ = ord (a) \cdot m

, we get

a^{∣ G ∣} = (a^{ord (a)})^{m} = e^{m} = e

. □

9 Fundamentals of Group Homomorphisms

We will study homomorphisms in depth later, but it is convenient to state the definition and basic properties now.

Definition 36 (Group homomorphism).

A map

φ : G \to G^{'}

between groups is called a group homomorphism if

φ (ab) = φ (a) φ (b) for all a, b \in G .

Theorem 37 (Basic properties of homomorphisms).

Let

φ : G \to G^{'}

be a group homomorphism. Then:

$φ (e_{G}) = e_{G^{'}}$ .
$φ (a^{- 1}) = φ (a)^{- 1}$ .
$φ (a^{n}) = φ (a)^{n}$ for all $n \in Z$ .

Proof.

(1) We have

φ (e_{G}) = φ (e_{G} \cdot e_{G}) = φ (e_{G}) φ (e_{G})

. Multiplying both sides by

φ (e_{G})^{- 1}

yields

e_{G^{'}} = φ (e_{G})

(2)

φ (a) φ (a^{- 1}) = φ (a a^{- 1}) = φ (e_{G}) = e_{G^{'}}

, so

φ (a^{- 1}) = φ (a)^{- 1}

(3) For

n \geq 0

, this follows by induction. For

n < 0

, use part (2). □

Definition 38 (Kernel and image).

For a group homomorphism

φ : G \to G^{'}

The kernel of $φ$ is $ker φ = {a \in G ∣ φ (a) = e_{G^{'}}}$ .
The image of $φ$ is $Im φ = {φ (a) ∣ a \in G}$ .

One can verify that

ker φ \leq G

and

Im φ \leq G^{'}

Theorem 39 (Injectivity and the kernel).

A group homomorphism

φ : G \to G^{'}

is injective if and only if

ker φ = {e_{G}}

Proof.

(\Rightarrow)

: If

φ (a) = e_{G^{'}} = φ (e_{G})

, then injectivity gives

a = e_{G}

(\Leftarrow)

: If

φ (a) = φ (b)

, then

φ (a b^{- 1}) = φ (a) φ (b)^{- 1} = e_{G^{'}}

, so

a b^{- 1} \in ker φ = {e_{G}}

, whence

a = b

. □

10 Summary and Next Steps

In this chapter we have systematically developed the foundations of group theory:

The group axioms and fundamental examples.
Uniqueness of the identity and inverses, cancellation laws, and solubility of equations.
Laws of exponents and the order of an element.
The definition of subgroups and criteria for recognizing them.
The structure of cyclic groups and their subgroups.
Group homomorphisms: definition, kernel, image, and the injectivity criterion.

Group Theory Algebra Textbook Group Definition Subgroups Cyclic Groups

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Groups: Definitions and First Properties

1 The Group Axioms

2 Fundamental Examples

3 Uniqueness of the Identity and Inverses

4 Cancellation Laws

5 Powers and Order of an Element

6 Subgroups

7 Generators and Cyclic Groups

8 Order of a Group

9 Fundamentals of Group Homomorphisms

10 Summary and Next Steps

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Groups: Definitions and First Properties

1 The Group Axioms

2 Fundamental Examples

3 Uniqueness of the Identity and Inverses

4 Cancellation Laws

5 Powers and Order of an Element

6 Subgroups

7 Generators and Cyclic Groups

8 Order of a Group

9 Fundamentals of Group Homomorphisms

10 Summary and Next Steps

Share your expertise with the world

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Cosets and Lagrange's Theorem

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