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...

In mathematics, a permutation of a set can mean one of two different things: an arrangement of its members in a sequence or linear order, or the act or...

establish properties of the group G. Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space...

1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered. Sometimes the...

n {\displaystyle n} factorial) such permutation operations, the order (number of elements) of the symmetric group S n {\displaystyle \mathrm {S} _{n}}...

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...

{\displaystyle G} as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two. A permutation representation...

which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents...

where Sn is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set ⁠ { −...

2. Such multiply transitive permutation groups can be defined for any natural number k. Specifically, a permutation group G acting on n points is k-transitive...

them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Suppose that E {\displaystyle E}...

used to obtain those positions, and so the group of symmetries of a square is isomorphic to the permutation group generated by (1234) and (13). The symmetries...

equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is by definition solvable by radicals if...

a subgroup of the symmetric group Sym ⁡ ( G ) {\displaystyle \operatorname {Sym} (G)} whose elements are the permutations of the underlying set of G....

in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as...

the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If...

other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations...

theory of permutation groups such as the order of an element of a group, conjugacy, and the cycle decomposition of elements of permutation groups. Ruffini...

mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly...

finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was...

A1(9) and to the derived group B2(2)′. A8 is isomorphic to A3(2). Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1...

M24 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier...

cyclic group C2. Sims (1973) proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory...

computational group theory, named after the mathematicians Otto Schreier and Charles Sims. This algorithm can find the order of a finite permutation group, determine...

respective permutations. The Rubik's Cube group is the subgroup of the symmetric group S 48 {\displaystyle S_{48}} generated by the six permutations corresponding...

(the other is the Janko group J3). It was constructed by Marshall Hall and David Wales (1968) as a rank 3 permutation group on 100 points. Both the Schur...

properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions...

memory permitting. Finite groups can be defined as groups of permutations and it is also possible to define finitely presented groups by specifying generators...

mathematical permutations. Alternating permutation Circular shift Cyclic permutation Derangement Even and odd permutations—see Parity of a permutation Josephus...

Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying...