# Category: Permutation groups

O'Nan–Scott theorem
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theor
Parker vector
In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements.
Covering groups of the alternating and symmetric groups
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symme
Schreier vector
In mathematics, especially the field of computational group theory, a Schreier vector is a tool for reducing the time and space complexity required to calculate orbits of a permutation group.
Schreier–Sims algorithm
The Schreier–Sims algorithm is an algorithm in computational group theory, named after the mathematicians Otto Schreier and Charles Sims. This algorithm can find the order of a finite permutation grou
Block (permutation group theory)
In mathematics and group theory, a block system for the action of a group G on a set X is a partition of X that is G-invariant. In terms of the associated equivalence relation on X, G-invariance means
Strong generating set
In abstract algebra, especially in the area of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a . A st
Multiply transitive group
No description available.
Wreath product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective f
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point. They are name
Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In p
Zassenhaus group
In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.
Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating gro
Automorphisms of the symmetric and alternating groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of stu
Jucys–Murphy element
In mathematics, the Jucys–Murphy elements in the group algebra of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: The
Generalized symmetric group
In mathematics, the generalized symmetric group is the wreath product of the cyclic group of order m and the symmetric group of order n.
Affine symmetric group
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensiona
Faro shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling playing cards, in which half of the deck is held in each hand with the thumbs inward, then cards are
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested
Multiply transitive group action
A group acts 2-transitively on a set if it acts transitively on the set of distinct ordered pairs . That is, assuming (without a real loss of generality) that acts on the left of , for each pair of pa
Hall's universal group
In algebra, Hall's universal group isa countable locally finite group, say U, which is uniquely characterized by the following properties. * Every finite group G admits a monomorphism to U. * All su
Jordan's theorem (symmetric group)
In finite group theory, Jordan's theorem states that if a primitive permutation group G is a subgroup of the symmetric group Sn and contains a p-cycle for some prime number p < n − 2, then G is either
System of imprimitivity
The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the
Primitive permutation group
In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into
Burnside ring
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the
Gassmann triple
In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a group G together with two faithful actions on sets X and Y, such that X and Y are not isomorphic as G-sets but every element of G has
Permutation representation
In mathematics, the term permutation representation of a (typically finite) group can refer to either of two closely related notions: a representation of as a group of permutations, or as a group of p
Sims conjecture
In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if is a primitive permutation group on a finite set and denotes the stabilizer
Rubik's Cube group
The Rubik's Cube group is a group that represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set corresponds to a cube move, which is the effect of any sequence of rotati
Base (group theory)
Let be a finite permutation group acting on a set . A sequence of k distinct elements of is a base for G if the only element of which fixes every pointwise is the identity element of . Bases and stron
List of transitive finite linear groups
In mathematics, especially in areas of abstract algebra and finite geometry, the list of transitive finite linear groups is an important classification of certain highly symmetric actions of finite gr