A **permutation group** is a mathematical structure consisting of a set of permutations that can be combined in a way that satisfies the properties of a group. Specifically, if you have a set \( X \), a permutation is a bijective function that rearranges the elements of \( X \). The collection of all possible permutations of a finite set \( X \) of size \( n \) is called the symmetric group, denoted as \( S_n \).
An **alternating group**, denoted \( A_n \), is a specific type of group in the field of abstract algebra. It consists of all the even permutations of a finite set of \( n \) elements. To fully understand this concept, it's important to break down a few terms: 1. **Permutation**: A permutation of a set is a rearrangement of its elements.
An automorphism of a group is an isomorphism from the group to itself. In the context of symmetric groups \( S_n \) and alternating groups \( A_n \), automorphisms play a significant role in understanding the structure and properties of these groups. ### Symmetric Groups \( S_n \) 1.
In the context of permutation group theory, a "block" is a concept related to the action of a group on a set.
The Burnside ring is a construction in algebra that arises in the study of group actions. Specifically, it is related to the representation theory of finite groups and has applications in combinatorics and algebraic topology. Given a finite group \( G \) acting on a set \( X \), the Burnside ring, denoted by \( \text{Br}(X, G) \), is formed by considering the isomorphism classes of finite \( G \)-sets.
Covering groups of the alternating group \( A_n \) and the symmetric group \( S_n \) are associated with the study of these groups in the context of their representations and the understanding of their structure. ### Symmetric Groups The symmetric group \( S_n \) consists of all permutations of \( n \) elements and has a very rich structure. Its covering groups can often be related to central extensions of the group.
A Faro shuffle, also known as a perfect shuffle, is a card shuffling method that interleaves two halves of a deck of cards in a precise manner. There are two types: the "in shuffle" and the "out shuffle." 1. **In Shuffle**: In this variation, the top card of the original deck remains in the top position after the shuffle.
A Frobenius group is a special type of group in group theory, which is a branch of mathematics. Specifically, a Frobenius group is a group \( G \) that satisfies certain properties related to its subgroups and the action of the group on a set.
The Gassmann triple refers to a specific concept in the field of geophysics and petrophysics, particularly in the study of the elastic properties of fluid-saturated rocks. It involves the characterization of the relationship between the bulk modulus, shear modulus, and density of a fluid-saturated porous rock.
The generalized symmetric group, usually denoted \( \text{GS}(n, k) \) or \( S(n, k) \), is a mathematical concept that generalizes the classical symmetric group, which consists of all permutations of a finite set. Specifically, the generalized symmetric group relates to the permutations and possible arrangements of \( n \) objects taken \( k \) at a time. ### Definition 1.
Hall's universal group, often denoted as \( H \), is a type of infinite group that arises in group theory, specifically in the context of group actions and representations. It is named after Philip Hall, who introduced it in the context of group theory. More specifically, Hall's universal group can be thought of as the group of finitely generated groups or, in a broader sense, the group of groups that allows one to categorize all groups that satisfy certain properties.
Jordan's theorem in the context of symmetric groups refers to a result concerning the structure of finite symmetric groups, \( S_n \). The theorem states that any transitive subgroup of \( S_n \) has a normal subgroup that is either abelian or contains a subgroup of index at most \( n \).
The list of transitive finite linear groups refers to a classification of finite groups that act transitively on a finite set and can be represented by matrices over a finite field. In the context of group theory, a linear group is a group of matrices that exhibits certain algebraic properties and is defined over a field (often a finite field).
In group theory, a branch of abstract algebra, the concept of **group action** describes how a group operates on a set. A group action can be defined mathematically, and it captures the essence of symmetry in algebraic structures.
The O'Nan–Scott theorem is a significant result in the field of group theory, particularly in the study of finite groups. It was formulated by John O'Nan and David Scott in the 1970s. The theorem provides a classification of the finite simple groups that can act as automorphism groups of certain types of groups, providing insight into the structure of finite groups and their representations.
A permutation group is a mathematical concept in group theory that consists of a set of permutations of a given set combined with the operation of composition. Here's a more detailed breakdown of the concept: 1. **Permutations**: A permutation of a set is a rearrangement of its elements. For a finite set with \( n \) elements, a permutation is simply a bijective function from the set to itself.
The Rubik's Cube group, in the context of group theory, is a mathematical structure that represents the set of all possible configurations (or states) of a Rubik's Cube and the operations (moves) that can be performed on it. This is an example of a finite group in abstract algebra. ### Key Concepts: 1. **Group Definition**: A group is a set equipped with an operation that satisfies four properties: closure, associativity, identity, and invertibility.
The concept of a "system of imprimitivity" comes from the field of group theory and is often used in the study of group actions. In the context of group actions on sets, a system of imprimitivity is a partition of a set that is invariant under the action of a group.
The Zassenhaus group, named after Hans Zassenhaus, is a specific type of group arising in the context of finite groups, particularly in relation to group theory and algebra. It is defined in terms of certain properties of the structure of groups. More precisely, the Zassenhaus group is often referred to in discussions of certain maximal subgroups of finite groups, particularly in relation to p-groups (groups where the order is a power of a prime) and their derived subgroups.
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