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 ▼ Group
Definition D23
Abelian group

An D21: Algebraic structure $G = (X, +)$ is an abelian group if and only if
 (1) $$\forall \, x, y \in X : x + y \in X$$ (2) $$\forall \, x, y, z \in X : (x + y) + z = x + (y + z)$$ (3) $$\exists \, 0_G \in X : \forall \, x \in X : 0_G + x = x + 0_G = x$$ (4) $$\forall \, x \in X : \exists \, {-} x \in X : -x + x = x + (- x) = 0_G$$ (5) $$\forall \, x, y \in X : x + y = y + x$$

An D21: Algebraic structure $G = (X, \times)$ is an abelian group if and only if
 (1) $$\forall \, x, y \in X : x y \in X$$ (2) $$\forall \, x, y, z \in X : (x y) z = x (y z)$$ (3) $$\exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x$$ (4) $$\forall \, x \in X : \exists \, x^{-1} \in X : x^{-1} x = x x^{-1} = 1_G$$ (5) $$\forall \, x, y \in X : x y = y x$$
Results
 ▶ Cyclic group is Abelian ▶ Left and right cosets coincide in Abelian group