Groups

Def:

A binary operation $*$ on a set $G$ , is a function $* : G \times G \to G$ . For any $a, b \in G$ , we shall write $a * b = * (a, b)$ .
A binary operation $*$ on a set $G$ is associative if for all $a, b, c \in G$ we have that $a * (b * c) = (a * b) * c$ .
If $*$ is a binary operation on set $G$ we say elements $a, b \in G$ commute if $a * b = b * a$ . We say $*$ (or $G$ ) is commutative if for all $a, b \in G$ , $a * b = b * a$ .

A group is an ordered pair $(G, *)$ where $G$ is a set and $*$ is a binary operation on $G$ satisfying the following axioms:

$*$ is associative
there exists an element $e \in G$ , called the identity of $G$ , such that for all $a \in G$ we have $a * e = e * a = a$ .
For each $a \in G$ there exists $a^{- 1} \in G$ , called the inverse of $a$ , such that $a * a^{- 1} = a^{- 1} * a = e$ .
The group $(G, *)$ is called abelian (or commutative) if $a * b = b * a$ for all $a, b \in G$ . We say that $G$ is a finite group if the set $G$ is finite.

Prop: Let $G$ be group, the following equivalent:

$G$ is abelian
if for all $a, b \in G$ , $(a b)^{2} = a^{2} b^{2}$
there exists $n \in N$ such that for all $a, b \in G$ , has that $(a b)^{k} = a^{k} b^{k}$ , for $k \in {n, n + 1, n + 2}$ .

Prop: If $G$ is a group under the operation $*$ , then:

the identity of $G$ is unique
for each $a \in G$ , $a^{- 1}$ is uniquely determined
$(a^{- 1})^{- 1} = a$ for all $a \in G$
$(a * b)^{- 1} = b^{- 1} * a^{- 1}$
for any $a_{1}, a_{2}, \dots, a_{n} \in G$ the value $a_{1} * a_{2} * \dots * a_{n}$ is independent of how the expression is bracketed. This is called the generalised associative laws

Prop: Let $G$ be a group and let $a, b\in G. The equations $a x = b$ and $y a = b$ have unique solutions for $x, y \in G$ . In particular, the left and the right cancellation laws hold in $G$ .

if $a u = a v$ , then $u = v$ , and
if $u b = v b$ , then $u = v$

Def: An element $e$ is called a right identity, if for all $x \in G$ , then $e * x = x$ , and an element $y$ associated to $x$ is called a right inverse of $x$ . Similarly for the left definitions

Prop: Let $G$ be a set and $*$ an associative binary operation on $G$ . Assume that $G$ has a right identity, and every element has a right inverse, then $(G, *)$ is a group.

Def: Let $G = {g_{1}, g_{2}, \dots, g_{n}}$ be a finite group with $g_{1} = e$ . The multiplication table or group tableof $G$ is the $n \times n$ matrix whose $i, j$ entry is the group element $g_{i} g_{j}$ .