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
▶ Abelian group
▶ Finite group
▶ Generated subgroup
▶ P-group
▶ Trivial group