Group

Definition

A group is a set $G$ and an operation $$*:G\times G\rightarrow G$$ , denoted $$(G,*:G\times G\rightarrow G)$$ but mathematicians are lazy so we just write $$(G,*)$$

Such that the following axioms hold:

Axioms

Words	Formal
$$\forall a,b,c\in G:[(a*b)*c=a*(b*c)]$$	$*$ is associative, because of this we may write $$a*b*c$$ unambiguously.
$$\exists e\in G\forall g\in G[e*g=g*e=g]$$	$*$ has an identity element
$$\forall g\in G\exists x\in G[xg=gx=e]$$	All elements of $G$ have an inverse element under $*$, that is

Important theorems

Identity is unique

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Proof:

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Assume there are two identity elements, $e$ and $e`$ with $$e\ne e`$$ .

That is both:

• $$\forall g\in G[e*g=g*e=g]$$
• $$\forall g\in G[e'*g=g*e'=g]$$

But then $$ee'=e$$ and also $$ee`=e'$$ thus we see $$e'=e$$ contradicting that they were different.

Now we know the identity is unique, so we can give it a symbol:

Group	Identity element
$(G,+)$ - additive notation $a+b$	We denote the identity $0$, so $$a+0=0+a=a$$
$(G,*)$ - multiplicative notation $ab$	We denote the identity $1$, so $$1a=a*1=a$$
$\text{GL}(n,F)$ - the General linear group (All $n\times n$ matrices of non-zero determinant)	We denote the identity by $Id,I,I_n$ or sometimes $Id_n$ that is $$AI=IA=A$$