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

Proof:

---

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.