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

algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and...

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

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 a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the...

classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or...

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...

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

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...

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

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination...

mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...

In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G...

1960 and 2004, that culminated in a complete classification of finite simple groups. Group theory has three main historical sources: number theory, the...

a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many...

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...

smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually...

the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple...

particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In particular, the quaternion group is a solvable group given by the group extension...

G. Every finite p-group is nilpotent. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer...

several constructions. Finite direct products of group schemes have a canonical group scheme structure. Given an action of one group scheme on another by...

mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology...

group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite...

finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite...

Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform...

product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German:...

The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton...

of functions. In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols...

of finite groups use characters of modular representations. Characters of irreducible representations encode many important properties of a group and...

mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The...