群の定義

$G$ is Set.
Defined a mapping.
$$G \times G \longrightarrow G$$ $$a \in G, b \in G \quad (a,b) \mapsto ab \in G $$ $G$ is Group. if $G$ fills following conditions.

1. 1. $(ab)c = a(bc) \quad (a,b,c \in G)$
2. 2. $\exists e \in G, \forall x \in G, ex = xe = x$
3. 3. $\forall x \in G, \exists x^{-1}, xx^{-1} = x^{-1}x = e$

${\rm \bf Remark:Identity\ element\ is\ unique}$
(Proof)
$e,e' \in G$ is identity element
$e=ee'=e'e=e'$

${\rm \bf Remark:Inverse\ element\ is\ unique}$
(Proof)
$a,b,c \in G, b,c$ is inverse element of $a$
$ab=e$
$ca=e$
$b=(ca)b=c(ab)=c$