ThmDex – An index of mathematical definitions, results, and conjectures.

Processing math: 100%

ThmDex – An index of mathematical definitions, results, and conjectures.

▼	Set of symbols
▼	Alphabet
▼	Deduction system
▼	Theory
▼	Zermelo-Fraenkel set theory
▼	Set
▼	Binary cartesian set product
▼	Binary relation
▼	Map
▼	Operation
▼	N-operation
▼	Binary operation
▼	Enclosed binary operation
▼	Groupoid
▼	Semigroup
▼	Monoid

Definition D22

Group

Formulation 1

An D21: Algebraic structure

$G = (X, +)$ is a group if and only if

(1)	$\begin{equation} \forall \, x, y \in X : x + y \in X \end{equation}$
(2)	$\begin{equation} \forall \, x, y, z \in X : (x + y) + z = x + (y + z) \end{equation}$
(3)	$\begin{equation} \exists \, 0_G \in X : \forall \, x \in X : 0_G + x = x + 0_G = x \end{equation}$
(4)	$\begin{equation} \forall \, x \in X : \exists \, {-} x \in X : -x + x = x + (- x) = 0_G \end{equation}$

Formulation 2

An D21: Algebraic structure

$G = (X, \times)$ is a group if and only if

(1)	$\begin{equation} \forall \, x, y \in X : x y \in X \end{equation}$
(2)	$\begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}$
(3)	$\begin{equation} \exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x \end{equation}$
(4)	$\begin{equation} \forall \, x \in X : \exists \, x^{-1} \in X : x^{-1} x = x x^{-1} = 1_G \end{equation}$

Children

▶	D23: Abelian group
▶	D1082: Finite group
▶	D1301: Generated subgroup
▶	D1568: P-group
▶	D1240: Trivial group