Definition:Group – ProofWiki
Definition
A group is a semigroup with an identity (that is, a monoid) in which every element has an inverse.
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 $\circ$.
The operation $\circ$ can be referred to as the group law.
Let $a, b \in G$ such that $ = a \circ b$.
Then $g$ is known as the product of $a$ and $b$.
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 use the notation $\gen {G, \circ}$ for $\struct {G, \circ}$.
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.
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 $-2$.
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 $0$.
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 $1 – x$.
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.
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 $a \circ b = a \ast \paren {e \ast b}$.
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.
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