Trait mathru::algebra::abstr::AbelianGroup [−][src]
An Abelian group is a commutative group.
A Group is a triple $(\mathbb{A}, \circ, e)$, composed by a set
$\mathbb{A}$ and a binary inner operation $\circ$ and the element $e \in \mathbb{A}$
Definition
\circ: \mathbb{A} \times \mathbb{A} \rightarrow \mathbb{A} , (x, y) \mapsto x \circ y
- Closure
$
\forall x, y \in \mathbb{A},: x \circ y \in \mathbb{A}$ - associativity
$\forall x, y, z \in \mathbb{A}$: $x \circ (y \circ z) = (x \circ y) \circ z$ 3. $e$ neutral element(identity)
$\forall x \in \mathbb{A}$: $x \circ e = e \circ x = x$ - Inverse element
$
x^{-1} \in \mathbb{A}: x^{-1} \circ x = x \circ x^{-1} = e$ - Commutativity
$
\forall x, y, \in \mathbb{A}: x \circ y = y \circ x$