Groups and Abelian Groups

Modern abstract algebra provides a rigorous framework for studying symmetry, structures, and operations. This guide explores algebraic systems, tracing a path through semigroups and monoids to the study of groups and abelian groups.

1. Algebraic Systems and Binary Operations

An algebraic system is built upon a non-empty set and one or more operations that act on its elements.

Binary Operations and Closure

Let $A$ be a non-empty set. A binary operation $*$ on $A$ is a mapping:

$\phi: A \times A \to A$

We write $a * b$ instead of $\phi(a, b)$ . The binary operation is closed on $A$ if:

$\forall a, b \in A, \quad a * b \in A \quad \text{(Closure Property)}$

Examples of Closure

The set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$ is closed under addition ( $+$ ) and multiplication ( $\cdot$ ), since the sum or product of any two natural numbers is always a natural number.
The set $\mathbb{N}$ is not closed under subtraction ( $-$ ) because , nor is it closed under division () because .

Algebraic Systems

An algebraic system (or algebraic structure) is represented as an ordered tuple $(A, *, \oplus, \dots)$ consisting of a non-empty set $A$ and one or more closed binary operations defined on $A$ .

$(\mathbb{N}, +)$ is an algebraic system.
$(\mathbb{Z}, +, -)$ is an algebraic system, where $\mathbb{Z}$ is the set of all integers.
$(\mathbb{R}, +, \cdot, -)$ is an algebraic system with multiple operations over the real numbers .

text

       [ Non-Empty Set: A ]  +  [ Binary Operation: * ]
                                |
                                v
               Is a * b in A for all a,b in A?
                 /                     \
               Yes                      No
               /                         \
              v                           v
     [ Closed Structure ]       [ Not an Algebraic System ]

Algebraic Properties

Let $(A, *)$ be an algebraic system:

Commutativity: The operation $*$ is commutative in $A$ if: $\forall a, b \in A, \quad a * b = b * a$
Associativity: The operation $*$ is associative in if:

Theorem: Uniqueness of the Identity Element

In any algebraic system $(A, *)$ , the identity element, if it exists, is unique.

Proof:

Suppose $e_1$ and $e_2$ are both identity elements in $A$ under the operation $*$ .

Since $e_2$ is an identity element: $e_1 * e_2 = e_1 \quad \text{(1)}$

Equating $(1)$ and $(2)$ yields:

$e_1 = e_2$

Thus, the identity element is unique. $\blacksquare$

2. Semigroups and Monoids

We can classify algebraic systems based on the axioms they satisfy.

text

                   [ Algebraic Systems ]
                             |
                     (Closure Property)
                             v
                      [ Semigroups ]
                             |
                   (Associative Property)
                             v
                       [ Monoids ]
                             |
                    (Identity Element)
                             v
                        [ Groups ]
                             |
                     (Inverse Elements)
                             v
                    [ Abelian Groups ]
                             |
                   (Commutative Property)

Semigroups

An algebraic system $(A, *)$ is a semigroup if:

The operation $*$ is closed on $A$ .
The operation $*$ is associative: $\forall a, b, c \in A, \quad (a * b) * c = a * (b * c)$

Examples:

$(\mathbb{N}, +)$ and $(\mathbb{N}, \cdot)$ are semigroups.
$(\mathbb{N}, -)$ is not a semigroup because it is neither closed nor associative: $(5 - 3) - 1 = 1 \neq 5 - (3 - 1) = 3$

Subsemigroups

Let $(A, *)$ be a semigroup and let $B \subseteq A$ . The algebraic system $(B, *)$ is a subsemigroup of $(A, *)$ if $B$ is closed under :

$\forall a, b \in B, \quad a * b \in B$

Example: Let $(\mathbb{N}, +)$ be a semigroup. The set of even positive integers $\mathbb{E} = \{2, 4, 6, \dots\}$ forms a subsemigroup $(\mathbb{E}, +)$ since the sum of two even numbers is always even.

Theorem: Direct Product of Semigroups

If $(S_1, *_1)$ and $(S_2, *_2)$ are semigroups, then their direct product is a semigroup under the component-wise operation defined by:

$(s_1', s_2') * (s_1'', s_2'') = (s_1' *_1 s_1'', s_2' *_2 s_2'')$

Proof:

Closure: Since $S_1$ and $S_2$ are semigroups, $s_1' *_1 s_1'' \in S_1$ and . Therefore, the output tuple lies in .

Since the operation is closed and associative, $(S_1 \times S_2, *)$ is a semigroup. $\blacksquare$

Monoids

An algebraic system $(A, *)$ is a monoid if it is a semigroup that contains an identity element $e \in A$ :

Closure: $\forall a, b \in A \implies a * b \in A$ .
Associativity: $\forall a, b, c \in A \implies (a * b) * c = a * (b * c)$ .

Examples:

$(\mathbb{N}_0, +)$ where $\mathbb{N}_0 = \{0, 1, 2, \dots\}$ is a monoid with identity element .

Submonoids

Let $(M, *)$ be a monoid with identity element $e$ . A subset $S \subseteq M$ forms a submonoid $(S, *)$ if:

$S$ is closed under $*$ .
The identity element $e \in S$ .

3. Foundations of Groups

A group is an algebraic structure that allows for the safe inversion of its operations.

Definition of a Group

An algebraic system $(G, *)$ is a group if it satisfies the following four axioms:

Closure: For all $a, b \in G$ , $a * b \in G$ .
Associativity: For all $a, b, c \in G$ , $(a$ .

Theorem: Uniqueness of the Inverse

In any group $(G, *)$ , the inverse of each element is unique.

Proof:

Let $a \in G$ and let $e$ be the identity element of $G$ . Suppose $b$ and $c$ are both inverses of $a$ .

Since $b$ is an inverse of $a$ : $a * b = e \quad \text{(1)}$
Since $c$ is an inverse of $a$ : $a * c = e \quad \text{(2)}$

We can write $b$ as: $b = b * e$

Substitute equation $(2)$ into this expression: $b = b * (a * c)$

By associativity: $b = (b * a) * c$

Since $b$ is an inverse of $a$ , $b * a = e$ : $b = e * c = c$

Thus, $b = c$ , proving the inverse of $a$ is unique. $\blacksquare$

Fundamental Group Theorems

Theorem: Cancellation Laws

In any group $(G, *)$ , the left and right cancellation laws hold:

Left Cancellation: $a * b = a * c \implies b = c$
Right Cancellation: $b * a = c * a \implies b = c$

Proof (Left Cancellation):

Assume $a * b = a * c$ . Since $G$ is a group, $a \in G$ has a unique inverse $a^{-1} \in G$ . Multiply both sides on the left by :

$a^{-1} * (a * b) = a^{-1} * (a * c)$

Apply the associative property:

$(a^{-1} * a) * b = (a^{-1} * a) * c$

Since $a^{-1} * a = e$ :

$e * b = e * c \implies b = c$

A similar process using right-multiplication proves the right cancellation law. $\blacksquare$

Theorem: Inverse of a Product (Socks-and-Shoes Property)

For any elements $a, b$ in a group $(G, *)$ , the inverse of their product is:

$(a * b)^{-1} = b^{-1} * a^{-1}$

Proof:

Let $x = a * b$ and $y = b^{-1} * a^{-1}$ . To prove that $y$ is the inverse of , we must show that and .

$\text{First product: } x * y = (a * b) * (b^{-1} * a^{-1})$

Apply the associative property:

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

Similarly:

$\text{Second product: } y * x = (b^{-1} * a^{-1}) * (a * b)$

Since $(a * b) * (b^{-1} * a^{-1}) = (b^{-1} * a^{-1}) * (a * b) = e$ , the unique inverse of is .

4. Abelian Groups and Algebraic Examples

An Abelian Group (or commutative group) is a group $(G, *)$ that satisfies the commutative property:

$\forall a, b \in G, \quad a * b = b * a$

Theorem: Condition for Commutativity

If every element of a group $G$ is its own inverse, then $G$ must be abelian.

Proof:

Let $(G, *)$ be a group where $x^{-1} = x$ for all $x \in G$ . Let $a, b \in G$ . Since , their product . Because every element is its own inverse:

$a^{-1} = a, \quad b^{-1} = b, \quad \text{and} \quad (a * b)^{-1} = a * b$

By the socks-and-shoes property:

$(a * b)^{-1} = b^{-1} * a^{-1}$

Substitute the self-inverse properties into this equation:

$a * b = b * a$

Thus, $G$ is abelian. $\blacksquare$

Theorem: Commutativity from Powers

If $(a * b)^2 = a^2 * b^2$ for all $a, b$ in a group , then is an abelian group.

Proof:

By definition, $x^2 = x * x$ . The given equation can be written as:

$(a * b) * (a * b) = (a * a) * (b * b)$

Using the associative law:

$a * (b * a) * b = a * (a * b) * b$

Apply the left cancellation law to remove $a$ :

$(b * a) * b = (a * b) * b$

Apply the right cancellation law to remove $b$ :

$b * a = a * b$

Since $b * a = a * b$ for all $a, b \in G$ , the group $G$ is abelian. $\blacksquare$

Concrete Algebraic Examples

Example 1: Rational Numbers under Custom Composition

Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Let $*$ be a binary operation defined by:

$a * b = \frac{ab}{2}, \quad \forall a, b \in \mathbb{Q}^+$

Show that $(\mathbb{Q}^+, *)$ is an abelian group.

Proof:

Closure: For any $a, b \in \mathbb{Q}^+$ , the product $ab$ is positive and rational, meaning $\frac{ab}{2}$ is also a positive rational number. Thus, .

Since all five axioms are satisfied, $(\mathbb{Q}^+, *)$ is an abelian group. $\blacksquare$

Example 2: Non-Singular Matrices (Non-Abelian Group)

Let $M$ be the set of all $n \times n$ non-singular matrices (where the determinant $|A| \neq 0$ ) with real entries. Let the binary operation be matrix multiplication ( $\cdot$ ).

Closure: The product of two non-singular matrices is also non-singular because $|AB| = |A| \cdot |B| \neq 0$ . Thus, $AB \in M$ .
Associativity: Matrix multiplication is associative: .

However, because matrix multiplication is generally non-commutative ( $AB \neq BA$ ), $(M, \cdot)$ is a non-abelian group (also known as the General Linear Group, $GL_n(\mathbb{R})$ ).

5. Finite Groups and Modulo Systems

The order of a group $G$ , denoted by $|G|$ or $o(G)$ , is the number of elements in the set $G$ . If the set is finite, $G$ is a finite group; otherwise, it is an infinite group.

Cayley Tables (Composition Tables)

For finite groups, we can represent binary operations using a Cayley table.

Example 1: Multiplicative Group $G = \{1, -1\}$

This is an abelian group of order $|G| = 2$ .

| $\cdot$ | $1$ | $-1$ | | :--- | :--- | :--- | | $1$ | $1$ | $-1$ | | $-1$ | $-1$ | |

Closure: All elements in the grid lie within the set $\{1, -1\}$ .
Identity: $1$ acts as the multiplicative identity.
Inverse: Each element is its own inverse ( $1^{-1} = 1$ , $(-1)^{-1} = -1$ ).

Example 2: Multiplicative Group of Cube Roots of Unity $G = \{1, \omega, \omega^2\}$

Here, $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$ . The order of the group is $|G| = 3$ .

| $\cdot$ | $1$ | $\omega$ | $\omega^2$ | | :--- | :--- | :--- | :--- | | $1$ | $1$ | $\omega$ | | | | | | | | | | | |

Identity: $1$ is the identity element.
Inverses: $1^{-1} = 1$ , $\omega^{-1} = \omega^2$ , and .

Modulo Arithmetic Systems

Addition Modulo $m$ ( $+_m$ )

Let $m$ be a positive integer. For any $a, b \in \mathbb{Z}$ :

$a +_m b = (a + b) \pmod m$

This returns the remainder when $(a + b)$ is divided by $m$ .

Multiplication Modulo $p$ ( $\times_p$ )

Let $p$ be a positive integer. For any $a, b \in \mathbb{Z}$ :

$a \times_p b = (ab) \pmod p$

This returns the remainder when the product $ab$ is divided by $p$ .

Theorem: Group Properties of Modulo Systems

The set $\mathbb{Z}_m = \{0, 1, 2, \dots, m-1\}$ forms an abelian group of order $m$ under addition modulo $m$ ().

Let's look at the Cayley tables for these structures.

Cayley Table for $(\mathbb{Z}_6, +_6)$

| $+_6$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Identity: $0$ is the identity element.
Inverses: The inverse of each element $x$ is $6-x$ for $x \neq 0$ , and the inverse of $0$ is .

Cayley Table for $(\mathbb{Z}_7^*, \times_7)$

| $\times_7$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Identity: $1$ is the identity element.
Inverses: $1^{-1}=1, \quad 2^{-1}=4, \quad 3^{-1}=5, \quad 4^{-1}=2, \quad 5^{-1}=3, \quad 6^{-1}=6$

Order of an Element

Let $(G, *)$ be a group, and let $a \in G$ . The order of an element $a$ , denoted by $|a|$ or $o(a)$ , is the smallest positive integer such that:

$a^n = \underbrace{a * a * \dots * a}_{n \text{ times}} = e$

If no such positive integer exists, the element is said to have infinite order.

Example: In the multiplicative group $G = \{1, -1, i, -i\}$ where the identity is $e = 1$ :

$1^1 = 1 \implies o(1) = 1$
$(-1)^2 = 1 \implies o(-1) = 2$

6. Subgroups and Structure Preservation

A non-empty subset of a group can inherit its properties and form a smaller group under the same operation.

Definition of a Subgroup

Let $(G, *)$ be a group. A non-empty subset $H \subseteq G$ is a subgroup of $G$ , denoted by $H \le G$ , if $(H, *)$ is itself a group under the same operation.

Every group $(G, *)$ of order $|G| > 1$ has at least two trivial subgroups:

The trivial subgroup containing only the identity: $(\{e\}, *)$
The group itself: $(G, *)$

Any other subgroup is called a proper subgroup.

Examples:

$(\mathbb{Z}, +)$ is a subgroup of $(\mathbb{Q}, +)$ , which in turn is a subgroup of $(\mathbb{R}, +)$ .
Let $G = \{1, -1, i, -i\}$ under multiplication. is a subgroup of because satisfies all group properties.

Theorem: The One-Step Subgroup Test

A non-empty subset $H$ of a group $(G, *)$ is a subgroup of $G$ if and only if:

$\forall a, b \in H, \quad a * b^{-1} \in H$

Proof:

Direction 1 ( $\implies$ ):

Assume $H$ is a subgroup of $G$ .

Since $H$ is a group, if $b \in H$ , then its inverse $b^{-1} \in H$ .
Since $H$ is closed under the operation $*$ , for any , we have .

Direction 2 ( $\impliedby$ ):

Assume $H$ is a non-empty subset of $G$ such that for all $a, b \in H \implies a * b^{-1} \in H$ . We must prove that satisfies all four group axioms:

Identity Element: Since $H$ is non-empty, there exists some element $x \in H$ . Let $a = x$ and $b = x$ . By the hypothesis: $x * x^{-1} \in H \implies e \in H$ Thus, contains the identity element.

Since all axioms are satisfied, $H$ is a subgroup of $G$ . $\blacksquare$

7. Homomorphisms and Isomorphisms

To compare different algebraic structures, we use mappings that preserve their underlying operations.

Homomorphism

Let $(G, *)$ and $(G', \oplus)$ be two groups. A function $f: G \to G'$ is a if:

$\forall a, b \in G, \quad f(a * b) = f(a) \oplus f(b)$

This means the mapping yields the same result whether you apply the operation before or after the function.

text

       [ Element a, b in G ] ------------ f ------------> [ f(a), f(b) in G' ]
                |                                                 |
            (Apply *)                                         (Apply ⊕)
                v                                                 v
           [ a * b ] ---------------- f ------------------> [ f(a) ⊕ f(b) ]

Isomorphism

If a homomorphism $f: G \to G'$ is also a bijection (one-to-one and onto), then $f$ is an isomorphism. If such a function exists, the groups $G$ and $G'$ are , denoted by:

$G \cong G'$

Isomorphic groups share the same algebraic structure and properties, differing only in the names of their elements.

Summary

Algebraic Systems: Non-empty sets with one or more closed binary operations.
The Algebraic Hierarchy:
- Semigroup: Closed and associative.
- Monoid: A semigroup with an identity element.
- Group: A monoid where every element has an inverse.
- Abelian Group: A group that also satisfies the commutative property.
Modulo Systems: $(\mathbb{Z}_m, +_m)$ is always an abelian group, while $(\mathbb{Z}_p^*, \times_p)$ forms an abelian group if and only if is a prime number.

Assessed progress

Groups and Abelian Groups

Key Concepts

Algebraic System

Semigroup and Monoid

Group

Abelian Group

Subgroup

Quick Check

Discrete Mathematical Structures

Discrete Mathematical Structures