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.
An **Abelian Lie group** is a type of Lie group in which the group operation is commutative. This means that for any two elements \( g \) and \( h \) in the group \( G \), the following property holds: \[ g \cdot h = h \cdot g \] where \( \cdot \) represents the group operation.
An Abelian group, also known as a commutative group, is a set equipped with a binary operation that satisfies certain properties. Specifically, a group \((G, *)\) is called Abelian if it satisfies the following criteria: 1. **Closure**: For all \(a, b \in G\), the result of the operation \(a * b\) is also in \(G\).
An "algebraically compact group" is a concept primarily found in the context of algebraic groups, a subject at the intersection of algebra and geometry. In broad terms, an **algebraic group** is a group that is also an algebraic variety, meaning it can be described by polynomial equations. These groups arise in various branches of mathematics, including number theory, algebraic geometry, and representation theory.
The Baer–Specker group, often denoted as \( BS \), is a classical example in the field of group theory, specifically in the study of torsion-free abelian groups. It is an important structure for various reasons, including its role in representation theory and its properties as a divisible group.
In group theory, a branch of abstract algebra, a **basic subgroup** typically refers to a subgroup that exhibits certain essential properties in the context of finite group theory, particularly in relation to p-groups and the Sylow theorems. However, it's important to clarify that the term "basic subgroup" is not standard across all texts and contexts and can have specific meanings depending on the area of interest.
The term "Butler Group" could refer to a few different things, depending on the context. One prominent reference is to the Butler Group in the context of technology and research. The Butler Group was a well-known IT research and advisory firm that provided insights into emerging technologies, trends, and market analysis for businesses. They focused on helping organizations understand and leverage technology effectively.
In the context of module theory, a **cotorsion group** refers to an abelian group (or more generally, a module) where every element is "cotorsion" in a certain sense.
A cyclic group is a type of group in which every element can be expressed as a power (or multiple) of a single element, known as a generator. In more formal terms, a group \( G \) is called cyclic if there exists an element \( g \in G \) such that every element \( a \in G \) can be written as \( g^n \) for some integer \( n \).
In the context of group theory, a **divisible group** is a particular type of abelian group (a group where the group operation is commutative) that satisfies a specific divisibility condition related to its elements.
An **elementary abelian group** is a specific type of group that is both abelian (commutative) and has a particular structure in which every non-identity element has an order of 2. This means that for every element \( g \) in the group, if \( g \neq e \) (where \( e \) is the identity element of the group), then \( g^2 = e \).
In group theory, a branch of abstract algebra, an essential subgroup is a specific type of subgroup that has particular relevance in the context of group actions and the structure of groups. A subgroup \( H \) of a group \( G \) is said to be essential in \( G \) if it intersects every nontrivial subgroup of \( G \).
The group of rational points on the unit circle refers to the set of points \( (x, y) \) on the unit circle defined by the equation \[ x^2 + y^2 = 1 \] where both \( x \) and \( y \) are rational numbers (numbers that can be expressed as fractions of integers). To describe the rational points on the unit circle, we can parameterize the unit circle using trigonometric functions or with rational parameterization.
In the context of abelian groups, the term "height" can refer to a couple of different concepts depending on the specific area of mathematics being considered, such as group theory or algebraic geometry. 1. **In Group Theory**: The height of an abelian group can refer to a measure of the complexity of the group, particularly when it comes to finitely generated abelian groups.
The Herbrand quotient is a concept from model theory and mathematical logic, particularly within the context of the study of formal systems and the properties of logical formulas. It generally pertains to measuring certain aspects of structures in formal theories, especially in relation to the notion of definability and algebraic properties of models. Specifically, the Herbrand quotient is defined in the context of Herbrand's theorem, which relates to the concept of Herbrand universes and Herbrand bases.
A **locally compact abelian group** is a type of mathematical structure that combines concepts from both topology and group theory. Here's a breakdown of what this term means: 1. **Group**: In mathematics, a group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements.
In group theory, a **locally cyclic group** is a type of group that is, in a certain sense, generated by its own elements in a cyclic manner. More formally, a group \( G \) is said to be locally cyclic if every finitely generated subgroup of \( G \) is cyclic. This means that for any finite set of elements from \( G \), the subgroup generated by those elements can be generated by a single element.
In the context of abelian groups, the term "norm" can refer to a couple of different concepts depending on the specific field of mathematics being discussed. One common usage, particularly in algebra and number theory, is the notion of a norm associated with a field extension or a number field.
A primary cyclic group is a specific type of cyclic group in the field of group theory, a branch of abstract algebra. A cyclic group is one that can be generated by a single element, meaning that every element of the group can be expressed as a power (or multiple) of this generator.
A Prüfer group, also known as a Prüfer \(p\)-group, is a type of abelian group that can be defined for a prime number \(p\).
Prüfer's Theorem refers to a couple of important results in the context of graph theory, particularly regarding trees. Here are the two main aspects of Prüfer's Theorem often discussed: 1. **Prüfer Code (or Prüfer Sequence)**: The theorem states that there is a one-to-one correspondence between labeled trees with \( n \) vertices and sequences of length \( n-2 \) made up of labels from \( 1 \) to \( n \).
In group theory, a "pure subgroup" refers to a specific type of subgroup within an abelian group. Specifically, a subgroup \( H \) of an abelian group \( G \) is called a **pure subgroup** if it satisfies a certain property concerning integer multiples.
The **rank** of an abelian group is a concept that generally refers to the maximum number of linearly independent elements in the group when it is considered as a module over the integers. For finitely generated abelian groups, the rank can be understood in relation to the structure theorem for finitely generated abelian groups.
A **topological abelian group** is a mathematical structure that combines the concepts of a group and a topology. Specifically, it is an abelian group that has a compatible topology, allowing for the notions of continuity and convergence to be defined in the context of group operations.
In the context of group theory, particularly in the study of abelian groups (and more generally, in the context of modules over a ring), the **torsion subgroup** is an important concept. The torsion subgroup of an abelian group \( G \) is defined as the set of elements in \( G \) that have finite order.
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