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...

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...

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 ⁠ { −...

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...

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...

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

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...

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

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...

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...

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...

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...

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}...

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...

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...

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....

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

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...

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

and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points. All the subquotients...

M22 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier...

both the permutation groups and the matrix groups. The upper bound on the order of G given by |G| ≤ 2N shows that G is finite. The black box groups were introduced...

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...

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...

(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...

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...

Let G {\displaystyle G} be a finite permutation group acting on a set Ω {\displaystyle \Omega } . A sequence B = [ β 1 , β 2 , . . . , β k ] {\displaystyle...

permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3...