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Abelian group theory is a branch of abstract algebra that focuses on the study of Abelian groups (or commutative groups). An **Abelian group** is a set equipped with an operation that satisfies certain properties: 1. **Closure**: For any two elements \( a \) and \( b \) in the group, the result of the operation (usually denoted as \( a + b \) or \( ab \)) is also in the group.

Algebraic groups are a central concept in an area of mathematics that blends algebra, geometry, and number theory. An algebraic group is defined as a group that is also an algebraic variety, meaning that its group operations (multiplication and inversion) can be described by polynomial equations. More formally, an algebraic group is a set that satisfies the group axioms (associativity, identity, and inverses) and is also equipped with a structure of an algebraic variety.

Combinatorial group theory is a branch of mathematics that studies groups by using combinatorial methods and techniques. It focuses on understanding the properties of groups through their presentations, generators, and relations. The main goal is to analyze and classify groups by examining how these elements can be combined and related in various ways.

Coxeter groups are abstract algebraic structures that arise in various areas of mathematics, including geometry, group theory, and combinatorics. They are defined by a particular type of presentation that involves reflections across hyperplanes in Euclidean space, but they can also be studied in a more abstract way.

Functional subgroups are specific categories or subdivisions within a larger organization or system that focus on a particular function or area of expertise. These subgroups are typically formed to enhance efficiency, streamline processes, and improve specialization in various tasks and responsibilities. For example, in a corporate setting, functional subgroups could include: 1. **Human Resources** - Focused on recruitment, employee relations, training, and benefits management.

In the context of group theory, an automorphism is an isomorphism from a group to itself. More formally, let \( G \) be a group. An automorphism is a function \( \phi: G \to G \) that satisfies the following properties: 1. **Homomorphism**: For all elements \( a, b \in G \), \( \phi(ab) = \phi(a) \phi(b) \).

"Group products" can refer to a couple of different concepts depending on the context, but it generally relates to a category of goods that are grouped together based on certain characteristics. Here are a few interpretations: 1. **Marketing and Retail**: In marketing, group products can refer to items that are sold together, often because they complement each other.

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.

Moonshine theory, also known simply as "moonshine," is a fascinating area of research in mathematics that explores deep connections between number theory, algebra, and mathematical physics. The term originally arises from the surprising mathematical phenomena discovered by John McKay in 1978 and further developed by others, including Richard Borcherds and Hollis Lang. At its core, moonshine refers to the conjectural relationships between finite groups and modular forms.

In mathematics, specifically in group theory, an **ordered group** is a group that is equipped with a total order that is compatible with the group operation. This means that not only do the group elements have a way to be compared (one element can be said to be less than, equal to, or greater than another), but the group operation respects this order.

P-groups, or *p-groups*, are a specific type of group in the field of abstract algebra, particularly in the study of group theory. A group \( G \) is classified as a p-group if the order (the number of elements) of the group is a power of a prime number \( p \). Formally, this can be expressed as: \[ |G| = p^n \] for some non-negative integer \( n \).

Representation theory of groups is a branch of mathematics that studies how groups can be represented through linear transformations of vector spaces. More formally, a representation of a group \( G \) is a homomorphism from \( G \) to the general linear group \( GL(V) \) of a vector space \( V \). This means that each element of the group is associated with a linear transformation, preserving the group structure.

Subgroup properties in group theory refer to certain characteristics or conditions that a subgroup of a given group may satisfy. These properties help in categorizing subgroups and understanding their structure relative to the larger group.

A **subgroup series** in group theory is a sequence of subgroups of a given group \( G \) that is organized such that each subgroup is a normal subgroup of the next one in the series.

Topological groups are a mathematical structure that combines concepts from both topology and group theory. Specifically, a topological group is a set equipped with two structures: a group structure and a topology, such that the group operations (multiplication and taking inverses) are continuous with respect to the topology.

In various contexts, a (B, N) pair can refer to different concepts depending on the field of study.

An **acylindrically hyperbolic group** is a type of group in geometric group theory that generalizes the concept of hyperbolic groups. These groups are characterized by a specific type of action they have on a $\textit{proper geodesic metric space}$.

The affine group is a mathematical concept that arises in the context of geometry and linear algebra. It is essentially a group that consists of affine transformations, which are a generalization of linear transformations that include translations.

The concept of an "approximate group" arises in the field of group theory and is particularly relevant in the study of discrete groups, geometric group theory, and number theory. An approximate group can be thought of as a structure that shares some properties with groups but does not necessarily satisfy all group axioms in a strict sense.

In group theory, the Artin transfer is a specific homomorphism associated with a certain class of groups called "finite groups." More specifically, it is related to the study of group extensions and the relationships between a group and its normal subgroups. The Artin transfer is particularly relevant in the context of modular representation theory and the representation theory of finite groups of Lie type, as well as in the study of central extensions and cohomology.