> [!NOTE] Definition (Finite Group & Order) > The [[Groups|group]] $(G,*)$ is said to be a *finite group* with *order* $|G|$ if $G$ is a [[Finite Set|finite set]]. > # Properties **Order of group elements**: The [[Order of Algebraic Structure|order]] of a group is the [[Cardinality|cardinality]] of its set. Note that [[Element of a finite group is of finite order|every element of a finite group has finite order]]. The [[Lagrange's theorem (on Finite Groups)|Lagrange theorem]] asserts that the order of every subgroup of a finite group divides the order of the finite group. A consequence of the theorem is that [[Order of Element of Finite Group Divides Order of The Group|order of any element of a finite group divides the order of the group]]. # Applications **Examples**: The [[Symmetric Groups of Finite Degree|the symmetric group of degree n]], denoted $S_{n},$ is an example of a finite group.