Group

A group $G$ is a set together with an operation $\cdot \,$(the dot is frequently suppressed, so $ab$ is written instead of $a\cdot b$) satisfying the following conditions, known as the group axioms:

• For all $a,b,c\in G$, $a(bc)=(ab)c$ (associativity).
• There exists an element $e\in G$ so that for all $g\in G$, $ge=eg=g$ (identity).
• For any $g\in G$, there exists $g^{-1}\in G$ so that $gg^{-1}=g^{-1}g=e$ ( inverses).

(Equivalently, a group is a monoid with inverses.)

Note that the group operation need not be commutative. If the group operation is commutative, we call the group an abelian group (after the Norwegian mathematician Niels Henrik Abel). Also, the group operation may be additative, with $+$ being used to display the operation.

Groups may be written as $(G,\cdot)$, G is a set and $\cdot$ is an operation. The order of $(G,\cdot)$ is the cardinality of G, the underlying set. The order of $(G,\cdot)$ is denoted as |$(G,\cdot)$|. (Note: It is common to abuse notation by instead writing $G$ instead of the full notation. We will use $G$ instead.) If |$G$| is finite, $G$ is a finite group. If not, $G$ is infinite.

Groups frequently arise as permutations or symmetries of collections of objects. For example, the rigid motions of $\mathbb{R}^2$ that fix a certain regular $n$-gon is a group, called the dihedral group and denoted in some texts $D_{2n}$ (since it has $2n$ elements) and in others $D_n$ (since it preserves a regular $n$-gon). Another example of a group is the symmetric group $S_n$ of all permutations of $\{1,2,\ldots,n\}$.