Group Theory for Physicists

2.2 Group and its Multiplication Table

Symmetric Transformation: A transformation is called the symmetric transformation of a system if it preserves the system invariant.

Definition of a Group:
A group $G$ is a set of symmetric transformation with a defined multiplication of elements, and satisfies the following rules:

  1. The set is closed to multiplication
    $$ RS\in G, \forall R,S\in G $$
  2. Associative
    $$ R(ST)=(RS)T $$
  3. Identical elements $$ \exists E\in G, ER=E, \forall R\in G $$
  4. Inverse elements $$ R^{-1} \in G, RR^{-1}=E,\forall R\in G $$

Commutative is not necessary. If the group is also commutative, it’s called Abelian.

The order of a finite group: The number of elements.

Continuos group: If the elements in an infinite group can be described by a set of continuos parameters, it’s called a continuos group.

$$ \forall R\in G, S\in G, R' \in G', S'\in G' $$$$ \exists R'\leftrightarrow R, S'\leftrightarrow S, R'S'\leftrightarrow RS $$

A complex of elements: A subset in the group $G$ is called a complex of elements.

Equality of two complexes: $R\subset S$ and $S\subset R$, then $R=S$

Product of two complexes:
$R={R_1,R_2,\dots,R_m}$, $S={S_1,S_2,\dots,S_n}$, $RS={R_jS_k}$, $j=1,\dots,m;k=1,\dots,n$, the product of two complexes is still a complex

$$ \begin{aligned} TG&=\{T,TR,TS,\dots\}\\ GT&=\{T,RT,ST,\dots\}\\ G^{-1}&=\{E,R^{-1},S^{-1},\dots\} \end{aligned} $$

$G=TG=GT=T^{-1}$

Proof
From the definition of group, $T\in G, TR\in G, TS\in G,\dots$, so $TG\subset G$;
Any $R$ in $G$ can be expressed by $R=T(T^{-1}R)$, $T^{-1}R\in G$, $R\in TG$, so $G\subset TG$

Multiplication table of a group: top$\leftrightarrow$right; left$\leftrightarrow$left;

From rearrangement theorem, there’s no repetitive elements in each row and in each column of the group table.

$$ C_N=\{E,R,R^2,\dots,R^{N-1}\}, R^N=E $$

$N$ is called the order of $C_N$, and $C_N$ is Abelian.

Proper rotation: pure rotation; Improper rotation: pure rotation followed by a space inversion.

The order of the elements: In Finite Group, lowerst $n$ for $R^n=E$ is called the order of the element $R$.

The period of $R$: The subset of powers of $R$ in $G$ is called the period of $R$.

Generators: The elements are called the generators of a finite group $G$ if each element in $G$ can be expressed as a multiplication of the generators, and each generator is not equal to the multiplication of other generators.

Rank of a group: The least number of generators.