Axiom:Group Axioms/Left
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Definition
A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:
\((\text G 0)\) | $:$ | Closure Axiom | \(\ds \forall a, b \in G:\) | \(\ds a \circ b \in G \) | |||||
\((\text G 1)\) | $:$ | Associativity Axiom | \(\ds \forall a, b, c \in G:\) | \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | |||||
\((\text G_{\text L} 2)\) | $:$ | Left Identity Axiom | \(\ds \exists e \in G: \forall a \in G:\) | \(\ds e \circ a = a \) | |||||
\((\text G_{\text L} 3)\) | $:$ | Left Inverse Axiom | \(\ds \forall a \in G: \exists b \in G:\) | \(\ds b \circ a = e \) |
Also see
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$