Groups

Subjects: Group Theory
Links: Operations and Structures

Def:

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

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

1. $\ast$ is associative
2. there exists an element $e\in G$, called the identity of $G$, such that for all $a\in G$ we have $a\ast e=e\ast a=a$.
3. For each $a\in G$ there exists $a^{-1}\in G$, called the inverse of $a$, such that $a\ast a^{-1}=a^{-1}\ast a=e$.
   The group $(G,\ast )$ is called abelian (or commutative) if $a\ast b=b\ast a$ 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 $a,b\in G$, $(ab{)}^2=a^2{b}^2$
• there exists $n\in \mathbb{N}$ such that for all $a,b\in G$, has that $(ab{)}^k=a^k{b}^k$, for $k\in \{n,n+1,n+2\}$.

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

• the identity of $G$ is unique
• for each $a\in G$, $a^{-1}$ is uniquely determined
• $(a^{-1}{)}^{-1}=a$ for all $a\in G$
• $(a\ast b{)}^{-1}=b^{-1}\ast a^{-1}$
• for any $a_1,a_2,\dots ,a_n\in G$ the value $a_1\ast a_2\ast \cdots \ast a_n$ 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 $ax=b$ and $ya=b$ have unique solutions for $x,y\in G$. In particular, the left and the right cancellation laws hold in $G$.

• if $au=av$, then $u=v$, and
• if $ub=vb$, then $u=v$

Def: An element $e$ is called a right identity, if for all $x\in G$, then $e\ast x=x$, 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 $\ast$ an associative binary operation on $G$. Assume that $G$ has a right identity, and every element has a right inverse, then $(G,\ast )$ 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 $g_i{g}_j$.