# Abelian Group ## Metadata **Status**:: #x **Zettel**:: #zettel/literature **Created**:: [[2024-03-27]] ## Synopsis Abelian group is a [[Group]] and the binary operator is commutative. - Abelian group is a set U under the binary operator ⨁. - ([[Magma]]) The binary operator is closed: $\forall a, b \in U: a \oplus b \in U$ - ([[Semigroup]]) The binary operator is associative: $(a \oplus b) \oplus c = a \oplus (b \oplus c)$ - ([[Monoid (Math)|Monoid]]) There exists an identity element: $\exists 0 \in U: a \oplus 0 = 0 \oplus a = a$ - ([[Group]]) Every element has an inverse: $\forall a \in U: \exists b \in U: a \oplus b = 0$ - (Abelian Group) The operator is commutative: $\forall a, b \in U: a \oplus b = b \oplus a$