In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or...

their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. An abelian group is a set A {\displaystyle A} , together...

the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter...

of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also...

classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the...

combinatorial group theory. A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said...

profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide...

generator of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic...

mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating...

permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves...