Group

A group $G$ is a set of elements together with an operation $\cdot:G\times G\to G$ (the dot is frequently supressed) satisfying the following conditions:

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).

Groups frequently arise as permutations 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 $D_{2n}$ (since it has $2n$ elements). Another example of a group is the symmetric group $S_n$ of all permutations of $\{1,2,\ldots,n\}$ .

Related algebraic structures are rings and fields.

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Group