A Group ((G,*))

A Group $(G,*)$

consists of

a set, $G$

and a binary operator, $*$

must

be associative, that is, if $a, b, c \in G$ then $(a*b) * c = a * (b * c)$

contain an identity, that is, $\exists e \in G$ s.t $e * a = a * e = a$ for each $a \in G$

contain an inverse for each element, that is, if $a \in G$ then $\exists a^\prime \in G$ s.t $a * a^\prime = a^\prime * a = e$

allows

cancellation, that is, for $a, b, c \in G$ then if $a * b = a * c$ or $b * a = c * a$ then $b = c$ # #

which is unique

might be

Abelian or commutative, that is, if $a, b \in G$ then $a * b = b * a$

e.g.

$(\mathbb{Z}, +)$

with identity $0$

non-Abelian

e.g.

$(\{\text{bijections on a set } S\}, \circ)$

with identify $\text{id}_S$