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A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005), no. 1, 154–183. The GAP Data Library "Primitive Permutation Groups". Carmichael, Robert D., Introduction to the Theory of Groups of Finite Order. Ginn, Boston, 1937. Reprinted by Dover Publications, New York, 1956. Todd Rowland. "Primitive Group Action ...
The eight O'Nan–Scott types of finite primitive permutation groups are as follows: HA (holomorph of an abelian group): These are the primitive groups which are subgroups of the affine general linear group AGL(d,p), for some prime p and positive integer d ≥ 1.
Pages in category "Permutation groups" The following 33 pages are in this category, out of 33 total. This list may not reflect recent changes. ...
Permutations without repetition on the left, with repetition to their right. If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which each element appears a number of times equal exactly to its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset ...
Considering the symmetric group S n of all permutations of the set {1, ..., n}, we can conclude that the map sgn: S n → {−1, 1} that assigns to every permutation its signature is a group homomorphism. [2] Furthermore, we see that the even permutations form a subgroup of S n. [1] This is the alternating group on n letters, denoted by A n. [3]
If X consists of n elements and G consists of all permutations, G is the symmetric group S n; in general, any permutation group G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.
The group M 24 is the permutation automorphism group of the extended binary Golay code W, i.e., the group of permutations on the 24 coordinates that map W to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.