Infinite group theory

Infinite group theory typically refers to the study of groups that are infinite in size, which can include a wide variety of mathematical structures in the field of abstract algebra. In mathematics, a group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses.

In group theory, the concept of commensurability relates to the way in which two groups can be compared based on their subgroups and certain structural properties. Two groups \( G \) and \( H \) are said to be **commensurable** if they share a common finite-index subgroup.

An **elementary amenable group** is a type of group in the field of group theory, specifically within the area of descriptive set theory and ergodic theory. Groups are classified as elementary amenable if they can be constructed from finite groups through a combination of certain operations.

An **FC-group** (or **Finite Class Group**) is a specific type of group in the field of group theory, a branch of mathematics. FC-groups are characterized by the property that every element has a finite number of conjugates, meaning that the set of conjugates for each element is finite.

A *free-by-cyclic group* is a specific type of group that can be thought of as a combination of two structures: a free group and a cyclic group. More formally, a free-by-cyclic group is a group \( G \) that can be expressed in the form: \[ G = F \rtimes C \] where \( F \) is a free group and \( C \) is a cyclic group.

Gromov's theorem on groups of polynomial growth states that any finitely generated group with polynomial growth is virtually nilpotent. This theorem is a significant result in geometric group theory and has important implications for the structure of groups.

Higman's embedding theorem is a result in the field of formal languages and automata theory, specifically relating to the study of recursively enumerable languages and context-free languages. The theorem provides a way to understand the structure of certain algebraic objects associated with these languages.

A Hopfian group is a type of group that satisfies a specific property related to its endomorphisms. Specifically, a group \( G \) is called a Hopfian group if every surjective (onto) endomorphism \( f: G \to G \) is an isomorphism.

The infinite dihedral group, usually denoted as \( D_{\infty} \) or sometimes \( D_{\infty}^* \), is a mathematical structure in group theory that extends the concept of the dihedral groups. While the finite dihedral group \( D_n \) represents the symmetries of a regular polygon with \( n \) sides (including rotations and reflections), the infinite dihedral group captures symmetries of an infinite linear arrangement.

In group theory, a branch of abstract algebra, an **infinite group** is a group that contains an infinite number of elements. In other words, if the cardinality (size) of the group is not a finite number, then the group is classified as infinite. Infinite groups can be categorized into various types based on their structure and properties.

A **locally finite group** is a type of group in the field of abstract algebra. Specifically, a group \( G \) is called locally finite if every finite subset of \( G \) generates a finite subgroup of \( G \). In other words, for any finite subset \( S \) of \( G \), the subgroup generated by \( S \), denoted by \( \langle S \rangle \), is finite.

A **Pro-p group** is a type of mathematical object in the field of group theory, particularly in the area of profinite groups. More specifically, a Pro-p group is a topological group that is both locally compact and totally disconnected, and it is defined as an inverse limit of finite groups whose orders are powers of a prime \( p \).

A **profinite group** is a type of topological group that has a very specific structure. These groups are characterized by several key features: 1. **Definition**: A profinite group is a compact, totally disconnected, Hausdorff topological group that is isomorphic to the inverse limit of a system of finite groups. In more intuitive terms, you can think of profinite groups as "limits" of finite groups that retain a group structure.

The Prüfer rank, also known as the Prüfer order, is a concept from the field of algebraic topology and algebraic K-theory that applies to modules, particularly in relation to Prüfer domains. It is a measure of the "size" of a module, similar to the rank of a vector space, but adapted for module theory.

A group \( G \) is called **residually finite** if for every nontrivial element \( g \in G \) (i.e., \( g \neq e \), where \( e \) is the identity element of the group), there exists a finite group \( H \) and a group homomorphism \( \varphi: G \to H \) such that \( \varphi(g) \neq \varphi(e) \).

The residue-class-wise affine group is a mathematical concept that arises in the context of group theory, specifically in relation to affine transformations and modular arithmetic. To understand it better, let's break down the terms involved: 1. **Affine Transformation**: An affine transformation can be viewed as a function that maps points from one vector space to another while preserving points, straight lines, and planes.

A Tarski monster group is a specific type of mathematical group that has some intriguing properties, particularly in the field of group theory. Specifically, a Tarski monster group is an infinite group in which every non-trivial subgroup has a prime order \( p \).

Thompson groups are a family of groups that arise in the area of geometric group theory, named after the mathematician J. G. Thompson who introduced them. They are defined in the context of homeomorphisms of the unit interval \([0, 1]\) and can be understood as groups of piecewise linear homeomorphisms.

Tits' alternative is a concept from group theory, named after mathematician Jacques Tits. It refers to a criterion for determining whether a given group is either "a linear group" or "a free group." More formally, it involves the classification of certain types of groups based on their actions on vector spaces.

The term "verbal subgroup" can refer to different concepts depending on the context, so it’s important to clarify where you might be encountering it. However, in mathematical group theory, a verbal subgroup typically refers to a subgroup generated by certain types of elements of a group that satisfy specific equations or properties.

The term "Z-group" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Mathematics**: In group theory, a Z-group could refer to a group that is isomorphic to the additive group of integers, denoted as \( \mathbb{Z} \). This is a fundamental example of an infinite cyclic group.