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A finite group is a mathematical structure consisting of a set with a finite number of elements and an operation that combines any two elements to form a third element, satisfying the properties of closure, associativity, the existence of an identity element, and the existence of inverse elements. finite groups are fundamental in abstract algebra and have applications in various fields, including physics, chemistry, and cryptography, due to their symmetry properties and structure.
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Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.
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Cayley's Theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G, meaning that any abstract group can be represented as a group of permutations. This theorem highlights the fundamental nature of permutation groups in group theory and provides a concrete way to understand abstract groups through permutation representations.
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Lagrange's Theorem in group theory states that for any finite group, the order (number of elements) of every subgroup divides the order of the group. This theorem provides a fundamental insight into the structure of groups, enabling deeper exploration of their properties and relationships.
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The symmetric group, denoted as S_n, is the group of all permutations of a finite set of n elements, capturing the essence of symmetry through its structure. It plays a fundamental role in group theory, serving as a cornerstone for understanding permutation groups and providing insights into the algebraic properties of symmetries in mathematics and physics.
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A permutation group is a mathematical concept that deals with the set of all possible rearrangements of a given set's elements, where the group operation is the composition of these rearrangements. It is a fundamental structure in abstract algebra, with applications in fields such as combinatorics, geometry, and the study of symmetry.
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A group homomorphism is a function between two groups that respects the group operation, meaning it maps the product of two elements in the first group to the product of their images in the second group. This structure-preserving map is fundamental in understanding how different groups relate to each other and forms the basis for many concepts in abstract algebra.
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Group isomorphism is a bijective homomorphism between two groups, indicating that they have the same structure in terms of group operations. This means that despite potential differences in the elements or representation, the groups are fundamentally the same in terms of their algebraic properties.
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A subgroup is a subset of a group that itself forms a group under the same operation as the original group. It must satisfy the group axioms: closure, associativity, identity, and invertibility within the subset.
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A normal subgroup is a subgroup that is invariant under conjugation by any element of the parent group, meaning it is preserved under the group's internal symmetries. This property ensures that the quotient group, formed by the parent group and the normal subgroup, is well-defined and itself a group.
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A quotient group, also known as a factor group, is formed by partitioning a group into disjoint subsets called cosets of a normal subgroup, and the operation on these cosets mirrors the group operation. This construction is fundamental in group theory as it allows the analysis of group structures by simplifying them into smaller, more manageable pieces.
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A group action is a formal way of interpreting the symmetries of a mathematical object by describing how each element of a group corresponds to a transformation of the object. This framework is crucial in understanding how groups operate on various structures, leading to insights in fields such as geometry, algebra, and topology.
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A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself, making it the building block for all finite groups through composition series. Understanding simple groups is crucial for group theory as they help classify all finite groups, akin to how prime numbers classify integers.
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An Abelian group is a set equipped with a binary operation that is associative, has an identity element, includes an inverse for each element, and is commutative. This structure is fundamental in abstract algebra and underpins many areas of mathematics, including number theory and topology.
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A non-Abelian group, also known as a non-commutative group, is a group in which the order of applying group operations matters, meaning that for some elements a and b in the group, the equation ab ≠ ba holds. These groups are fundamental in many areas of mathematics and physics, particularly in the study of symmetries and quantum mechanics, where they describe more complex structures than Abelian groups.
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Group representation is a mathematical concept where elements of a group are represented as matrices, which allows for the study of group properties through linear algebra. This approach is crucial in various fields, including physics and chemistry, as it facilitates the understanding of symmetry and invariance in systems.
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A cyclic group is a group that can be generated by a single element, meaning every element in the group can be expressed as a power of this generator. cyclic groups are always abelian, and they can be finite or infinite, with their structure being closely tied to the properties of integers under addition or multiplication modulo n.
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A composition series is a finite sequence of subgroups in a group such that each is a normal subgroup of the previous one, and the quotient groups are simple. This series provides a way to analyze the structure of a group by breaking it down into simpler, well-understood components.
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A finite simple group is a group that is both finite in size and simple, meaning it has no normal subgroups other than the trivial group and itself. These groups serve as the building blocks for all finite groups, analogous to prime numbers in the context of integers, and are completely classified by the monumental work known as the classification of finite simple groups.
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Alternating groups are the group of even permutations of a finite set and are important in the study of symmetric groups and group theory. They are simple for n ≥ 5, meaning they have no nontrivial normal subgroups, which makes them fundamental in the classification of finite simple groups.
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Finite simple groups are the building blocks of all finite groups, analogous to prime numbers in number theory, and they are fully classified by the monumental achievement known as the Classification of Finite simple groups. This classification consists of several infinite families and 26 exceptional groups known as the sporadic groups, with the Monster group being the largest among them.
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The order of an element in a group is the smallest positive integer n such that the n-th power of the element is the identity element of the group. This concept is fundamental in group theory, providing insights into the structure and behavior of algebraic systems.
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A spherical Coxeter group is a group generated by reflections across hyperplanes in a Euclidean space, where the group acts on a sphere and the corresponding Coxeter diagram is finite. These groups are fundamental in the study of symmetry and geometry, as they classify regular polytopes and tessellations in spherical spaces.
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A cyclic group is a group that can be generated by a single element, where every element in the group can be expressed as a power of this generator. These groups are fundamental in abstract algebra because they offer simple structures that serve as building blocks for more complex groups and have applications in number theory and cryptography.
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A reflection group is a group generated by reflections, which are isometries of a Euclidean space that fix a hyperplane pointwise. These groups play a crucial role in the study of symmetries and are foundational in understanding the geometric structure of spaces, particularly in the classification of finite groups and the theory of Coxeter groups.
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Fischer groups are a set of sporadic simple groups discovered by Bernd Fischer in the late 1960s, known for their intricate structures and roles within the classification of finite simple groups. These groups, specifically the three Fischer groups Fi22, Fi23, and Fi24, are among the 26 sporadic groups and are characterized by their unique properties and symmetry relations.
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Mathieu groups are a set of five sporadic simple groups that were among the first discovered and are known for their exceptional symmetry properties. They are named after Émile Mathieu, who discovered them in the 19th century, and they play a significant role in the classification of finite simple groups.
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In mathematics, a group is a set equipped with a single binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. Groups are central to abstract algebra and provide a foundational framework for studying symmetry, structure, and transformations across various mathematical disciplines.
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The order of a group in abstract algebra is the total number of elements in the group, which provides a fundamental property that influences the group's structure and behavior. It plays a crucial role in determining the possible subgroups and the applicability of various theorems, such as Lagrange's theorem, which states that the order of any subgroup divides the order of the group.
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The rank of a group in mathematics refers to the smallest number of elements required to generate the group, essentially measuring the group's complexity in terms of its generating set. It provides a fundamental insight into the structure and classification of groups, distinguishing between finite and infinite groups based on their generating capabilities.
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