Groups

A group $\mathcal{G}$ is a set of distinct elements $G_1,\cdots ,G_g$ s.t. for any two elements $G_i$ and $G_j$ there is a binary operation $\ast$ called group multiplication that satisfies the following four axioms

1. The set $\mathcal{G}$ is closed under multiplication.
   • $G_i,G_j$ $\in \mathcal{G}\to G_i\ast G_j$ $\in \mathcal{G}$
2. Associativity
   • $G_k\ast (G_i\ast G_j)=(G_k\ast G_i)\ast G_j$
3. $\exists$ identity $\in \mathcal{G}$
   • $E\ast G_i=G_i$
4. $\exists$ inverse $\forall G\in \mathcal{G}$
   • $G_i^{-1}{G}_i=E$

The total number of elements of a group is called the order of the group. The order can be either finite or infinite.

A group is said to be Abelian if the group elements commute

5. Commutativity $\forall G_i,G_j\in \mathcal{G}$
   • $G_i{G}_j=G_j{G}_i$

otherwise it is non-Abelian.

Let $G$ be an element of the group $\mathcal{G}$, the smallest integer $p$ such that $G^p=E$ is called the order of element $G$.

Multiplication Tables

Representations

Consider the abstract elements $G$ of a group $\mathcal{G}$, and another set of elements $\hat{G}$ in a group $\hat{\mathcal{G}}$. If every element $G$ has has a one to one correspondence with $\hat{G}$ then the groups $\hat{\mathcal{G}}$ and $\mathcal{G}$ are isomorphic, then $\hat{\mathcal{G}}$ is called a representation of $\mathcal{G}$.