Definition:Group

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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

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