General definitions

We take the above four properties of the symmetric group as the four defining properties of a group:

Definition 5.3.1   Let $G$ be a set and suppose that there is a mapping

$$\begin{array}{c} * : G\times G \longrightarrow G \\ \ \ \ \ (g_1,g_2) \longmapsto g_1*g_2 \end{array}$$

(called the group's operation) satisfying

(G1)

if $g_1, g_2$ belong to $G$ then $g_1*g_2$ belongs to $G$ (``$G$ is closed under $*$"),

(G2)

if $g_1, g_2, g_3$ belong to $G$ then $(g_1*g_2)*g_3 = g_1*(g_2*g_3)$ (``associativity"),

(G3)

there is an element $1 \in G$ such that $1*g = g*1 = g$ for all $g\in G$ (``existence of an identity"),

(G4)

if $g$ belongs to $G$ then there is an element $g^{-1} \in G$, called the inverse of $g$ such that $g*g^{-1} = g^{-1}*g = 1$ (``existence of inverse").

Then $G$ (along with the operation $*$) is a group.

In the above definition, we have not assumed that there was exactly one identity element $1$ of $G$ because, in fact, one can show that if there is one then it is unique. (To do this you can use the cancellation law: if $a*c=b*c$, where $a,b,c\in G$, then $a=b$.) Likewise, if $G$ is a group and $g\in G$ then the inverse element of $g$ is unique. (Prove it!) There are other properties of a group which can be derived from (G1)-(G4). We shall prove them as needed. The multiplication table ^5.1 of a finite group $G$ is a tabulation of the values of the binary operation $*$. This is also called the Cayley table, after the mathematician Arthur Cayley (1821-1895) who first introduced it in 1854 (along with the definition of an abstract group, as in Definition 5.3.1). Let $G=\{g_1,...,g_n\}$. The multiplication table of $G$ is:

*	$g_1$	$g_2$	...	$g_j$	...	$g_n$
$g_1$
$g_2$
$\vdots$
$g_i$				$g_i*g_j$
$\vdots$
$g_n$

Some properties:

Lemma 5.3.2 (a)   Each element $g_k\in G$ occurs exactly once in each row of the table. (b) Each element $g_k\in G$ occurs exactly once in each column of the table. (c) If the $(i,j)^{th}$ entry of the table is equal to the $(j,i)^{th}$ entry then $g_i*g_j=g_j*g_i$. (d) If the table is symmetric about the diagonal then $g*h=h*g$ for all $g,h\in G$. (In this case, we call $G$ abelian.)

Exercise 5.3.3   Compute the multiplication table for $C_{3}$.

Exercise 5.3.4   Compute the multiplication table for

$$\{ \left( \begin{array}{cc} 1 &0 \\ 0 &1 \end{array}... ...egin{array}{cc} -1 &0 \\ 0 &-1 \end{array} \right) \}.$$

Exercise 5.3.5   Compute the multiplication table for $C_{2}\times C_2$ (multiplication is componentwise).

Exercise 5.3.6   Compute the addition table for $\mathbb{Z}/4\mathbb{Z}$ (= $C_4$).

Exercise 5.3.7   Consider the mappings from $\mathbb{C}-\{0,1\}$ (all complex numbers except 0 and $1$) to itself given by $f_1(x)=x$, $f_2(x)=1/x$, $f_3(x)=1-x$, $f_4(x)=f_3(f_2(x))$, $f_5(x)=f_2(f_3(x))$, $f_6(x)=f_2(f_4(x))$. Let the group operation on $G=\{f_1, f_2, f_3, f_4, f_5, f_6\}$ be composition of functions. Show $G$ is a group. Write the Cayley table for $G$.