Groups

Definition
Let G be a set and $ \phi:X\times X\rightarrow X $ a binary operation.
Suppose $ \phi $ has properties
$ (i)\forall x,y,z \in G \phi(\phi(x,y),z)=\phi(x,\phi(y,z)) $
$ (ii) \exists \:e\in G \forall x\in G,\: \phi(x,e)=\phi(e,x)=x; $
$ (iii) \forall x\in G, \exists x'\in G\: \phi(x,x')=\phi(x',x)=e. $
Then couple $ (G,\phi) $ is called group.
If
$ (iv) \forall x,y \in G\: \phi(x,y)=\phi(y,x) $
$ (G,\phi)  $ is called a commutative group.
For a more simple notation we write $ x*y=\phi(x,y) $
Then group axioms become
$ (i)\forall x,y,z \in G \: (x*y)*z=x*(y*z); $
$ (ii) \exists \:e\in G \forall x\in G,\:x*e=e*x=x; $
$ (iii) \forall x\in G, \exists x'\in G\: x*x'=x'*x=e; $

Commutativity
$ (iv) \forall x,y \in G\:x*y=y*x; $
Group morphism
Let $ (G_1,\phi_1),(G_2,\phi_2) $ and $ f:G_1\rightarrow G_2 $ a function.
f is called morphism of groups if $ \forall x,y\in G_1,\:f(\phi_1(x,y))=\phi_2(f(x),f(y)) $
or with the other notation
$ f(x*_1y)=f(x)*_2f(y) $

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