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:
- The set is closed to multiplication
$$ RS\in G, \forall R,S\in G $$ - Associative
$$ R(ST)=(RS)T $$ - Identical elements $$ \exists E\in G, ER=E, \forall R\in G $$
- 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.
Isomorphic: If there’s a one-to-one correspondence between elements of two groups, and the correspondence is invariant to multiplication, the two groups are called isomorphic.
$$ \forall R\in G, S\in G, R' \in G', S'\in G' $$$$ \exists R'\leftrightarrow R, S'\leftrightarrow S, R'S'\leftrightarrow RS $$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
Rearrangement theorem: For any elements $T$ in a group, $G={E,R,S,\dots}$, the following three sets are all the same as the original group $G$:
$$ \begin{aligned} TG&=\{T,TR,TS,\dots\}\\ GT&=\{T,RT,ST,\dots\}\\ G^{-1}&=\{E,R^{-1},S^{-1},\dots\} \end{aligned} $$$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.
Cyclic group: A finite group constructed by the powers of one element $R$ is called the cyclic group, denoted by $C_N$.
$$ C_N=\{E,R,R^2,\dots,R^{N-1}\}, R^N=E $$$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.