theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on...

aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence...

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...

{\displaystyle P} . In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n {\displaystyle n} is...

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 group theory, a discipline within abstract algebra, a special group is a finite group of prime power order that is either elementary abelian itself...

complete classification of finite simple groups. Group theory has three main historical sources: number theory, the theory of algebraic equations, and...

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 (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic...

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...

representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which...

In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups...

In the theory of algebraic groups, a special group is a linear algebraic group G with the property that every principal G-bundle is locally trivial in...

automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The...

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...

known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose...

study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated...

of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field...

representation theory; this special case has very different properties. See Representation theory of finite groups. Compact groups or locally compact groups — Many...

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...

quotient space is finite. crystallographic point group congruence subgroup arithmetic group geometric group theory computational group theory freely discontinuous...

Minkowski's theorem Topological group Field Finite field Galois theory Grothendieck group Group ring Group with operators Heap Linear algebra Magma Module...

In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same...

alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...

FC-group A group is an FC-group if every conjugacy class of its elements has finite cardinality. finite group A finite group is a group of finite order, that...

one can prove that the center of any non-trivial finite p-group is non-trivial. If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G)...

finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups,...

affine group of an affine space over a finite field of prime order. Groups similar to Galois groups are (today) called permutation groups. The theory of permutation...

model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley...

field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme...