ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
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▼ Zermelo-Fraenkel set theory
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▼ Groupoid
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▼ Group
Definition D23
Abelian group
Formulation 0
An D21: Algebraic structure $G = (X, +)$ is an abelian 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}
(5) \begin{equation} \forall \, x, y \in X : x + y = y + x \end{equation}
Formulation 1
An D21: Algebraic structure $G = (X, \times)$ is an abelian 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}
(5) \begin{equation} \forall \, x, y \in X : x y = y x \end{equation}
Results
▶ R893: Cyclic group is Abelian
▶ R738: Left and right cosets coincide in Abelian group