# Category: Infinite group theory

Verbal subgroup
In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements fo
Infinite dihedral group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represe
Stable group
In 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 rank (see below).
Prüfer rank
In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections. The ran
Prüfer theorems
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Gromov's theorem on groups of polynomial growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nil
Prüfer group
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th r
Ulm's theorem
No description available.
Z-group
In mathematics, especially in the area of algebra 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
Growth rate (group theory)
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written
Elementary amenable group
In mathematics, a group is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in amenable groups when applied to amen
Residual property (mathematics)
In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X". Formally, a group G is residually X if for
Subgroup growth
In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let be a finitely generated group. Then, for each integer define to b
Hopfian group
In mathematics, a Hopfian group is a group G for which every epimorphism G → G is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Linear group
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a m
FC-group
In mathematics, in the field of group theory, an FC-group is a group in which every conjugacy class of elements has finite cardinality. The following are some facts about FC-groups: * Every finite gr
Basic subgroup
In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulik
Pro-p group
In mathematics, a pro-p group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a p-group. Note that, as profinite groups are compact, the ope
Infinite conjugacy class property
In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite. The von Neumann group
In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal nilpotent subgroups of the group. It was named by after Kurt Hirsch and Boris
Infinite group
In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.
Locally finite group
In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroup
Tarski monster group
In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup,
Height (abelian group)
In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solut
Free-by-cyclic group
In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group is said to be free-by-cyclic if it has a free normal
Thompson groups
In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , that were introduced by Richard Thompson in some unpublish
Hall's universal group
In algebra, Hall's universal group isa countable locally finite group, say U, which is uniquely characterized by the following properties. * Every finite group G admits a monomorphism to U. * All su
Residue-class-wise affine group
In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on (the integers), whose elements are bijectiveresidue-class-wise affine mappings. A
Higman's embedding theorem
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Gra
Profinite group
In mathematics, a 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 a "uniform", or "synopti
Residually finite group
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 there is a homomorphism h from G to a finite gro
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is
Commensurability (group theory)
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to t
Tame group
In mathematical group theory, a tame group is a certain kind of group defined in model theory. Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed fie