Finite group - Knowlpedia

A finite group is a group $G$ whose underlying set is a finite set. Its order, denoted $|G|$ , is the number of its elements. In particular, the trivial group has order $1$ .

Examples and classification

The additive group $\mathbb Z/n\mathbb Z$ has order $n$ , for every integer $n\geq1$ . The symmetric group $S_n$ has order $n!$ .

The classification of finite simple groups applies to the finite groups having no proper nontrivial normal subgroup and having order greater than one. General finite groups need not be simple; the finite-group table includes both kinds.

References

P. J. Cameron, Group Theory revision notes, §1.1, p. 2: finite groups and their orders.