01-01. Basic Axioms and examples
Definition.
- $G$,
- set
(1) A binary operation $\star$ on a set $G$ is a funciton $\star: G \times G \rightarrow G$. We will write $a \star b := \star(a, b)$ for any $a, b \in G$.
(2) A binary operation $\star$ is said to be associative if for all $a, b, c \in G$,
\[(a \star b) \star c = a \star (b \star c) .\](3) A binary operation $\star$ is said to be commutative if
\[\forall a, b \in G, \ a \star b = b \star a .\](4) $a, b$ is said to commute if $a \star b = b \star a$.
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Examples
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Definition. group
- $G$,
- set
(1) an ordred pair $(G, \star)$ is said to be group if
- (i) $(a \star b) c = a \star (b \star c)$ for all $a, b, c \in G$.
- (ii) there exists $e \in G$ such that $a \star e = e \start a = a$ for all $a \in G$.
- (iii) for each $a \in G$, tehre is an element $a^{-1} \in G$ such that $a \star a^{-1} = a^{-1} \star a = e$.
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