Groups

Posted November 25th, 2007 by Structure

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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