# Definition:Group

## Definition

A **group** is a semigroup with an identity (that is, a monoid) in which every element has an inverse.

### Group Axioms

The properties that define a **group** are sufficiently important that they are often separated from their use in defining semigroups, monoids, and so on, and given recognition in their own right.

A **group** is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:

\((\text G 0)\) | $:$ | Closure | \(\ds \forall a, b \in G:\) | \(\ds a \circ b \in G \) | |||||

\((\text G 1)\) | $:$ | Associativity | \(\ds \forall a, b, c \in G:\) | \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | |||||

\((\text G 2)\) | $:$ | Identity | \(\ds \exists e \in G: \forall a \in G:\) | \(\ds e \circ a = a = a \circ e \) | |||||

\((\text G 3)\) | $:$ | Inverse | \(\ds \forall a \in G: \exists b \in G:\) | \(\ds a \circ b = e = b \circ a \) |

These four stipulations are called the **group axioms**.

The notation $\struct {G, \circ}$ is used to represent a **group** whose underlying set is $G$ and whose operation is $\circ$.

### Group Law

The operation $\circ$ can be referred to as the **group law**.

### Product Element

Let $a, b \in G$ such that $ = a \circ b$.

Then $g$ is known as the **product of $a$ and $b$**.

### Multiplicative Notation

When discussing a general **group** with a general **group law**, it is customary to dispense with a symbol for this operation and merely concatenate the elements to indicate the product element.

- $x y$ is used to indicate the result of the operation on $x$ and $y$. There is no symbol used to define the operation itself.

- $e$ or $1$ is used for the identity element.

- $x^{-1}$ is used for the inverse element.

- $x^n$ is used to indicate the $n$th power of $x$.

Compare with additive notation.

## Also denoted as

Some sources denote a **group** using the notation $\gen {G, \circ}$ for $\struct {G, \circ}$.

## Examples

### Example: $\dfrac {x + y} {1 + x y}$

Let $G = \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.

Let $\circ: G \times G \to \R$ be the binary operation defined as:

- $\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

The algebraic structure $\struct {G, \circ}$ is a group.

### Example: $x + y + 2$ over $\R$

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

- $\forall x, y \in \R: x \circ y := x + y + 2$

Then $\struct {\R, \circ}$ is a group whose identity is $-2$.

### Example: $x + y + x y$ over $\R \setminus \set {-1}$

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

- $\forall x, y \in \R: x \circ y := x + y + x y$

Let:

- $\R' := \R \setminus \set {-1}$

that is, the set of real numbers without $-1$.

Then $\struct {\R', \circ}$ is a group whose identity is $0$.

### Example: $x^{-1} = 1 - x$

Let $S = \set {x \in \R: 0 < x < 1}$.

Then an operation $\circ$ can be found such that $\struct {S, \circ}$ is a group such that the inverse of $x \in S$ is $1 - x$.

### Example: Group of Linear Functions

Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as:

- $\forall x \in \R: \map {\theta_{a, b} } x = a x + b$

where $a, b \in \R$ such that $a \ne 0$.

The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group.

$\struct {G, \circ}$ is specifically non-abelian.

### Example: Operation Induced by Self-Inverse and Cancellable Elements

Let $S$ be a set with an operation which assigns to each $\tuple {a, b} \in S \times S$ an element $a \ast b \in S$ such that:

- $(1): \quad \exists e \in S: a \ast b = e \iff a = b$
- $(2): \quad \forall a, b, c \in S: \paren {a \ast c} \ast \paren {b \ast c} = a \ast b$

Then $\struct {S, \circ}$ is a group, where $\circ$ is defined as $a \circ b = a \ast \paren {e \ast b}$.

## Examples of Structures which are not Groups

### Arbitrary Example: Order 4

Let $S = \set {1, 2, 3, 4}$.

Consider the algebraic structure $\struct {S, \circ}$ given by the Cayley table:

- $\begin{array}{r|rrrr} \circ & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 3 & 1 \\ 3 & 3 & 2 & 4 & 3 \\ 4 & 4 & 3 & 1 & 2 \\ \end{array}$

Then $\struct {S, \circ}$ is not a group.

## Also see

- Results about
**groups**can be found**here**.

## Historical Note

The term **group** was first used by Évariste Galois in $1832$, in the context of the solutions of polynomials in radicals. Augustin Louis Cauchy was also involved in this development.

The concept of the group as a purely abstract structure was introduced by Arthur Cayley in his $1854$ paper *On the theory of groups*.

The first one to formulate the set of axioms to define the structure of a group was Leopold Kronecker in $1870$.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: Definition $1$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups: Definition $3$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.2$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26$ - 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Definition $1.1 \text{(iii)}$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $2.2$ Definitions $\text{(ii)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 33$. The definition of a group - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms - 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (next): $\S 1.1$: Definition $1.1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**group** - 1999: J.C. Rosales and P.A. García-Sánchez:
*Finitely Generated Commutative Monoids*... (previous) ... (next): Chapter $1$: Basic Definitions and Results - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$: Exercise $5$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**group** - 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Definition $1.4$