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The degree of a group of permutations of a finite set is the number of elements in the set. The order of a group (of any type) is the number of elements (cardinality) in the group. By Lagrange's theorem, the order of any finite permutation group of degree n must divide n! since n-factorial is the order of the symmetric group S n.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
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 ...
If is a permutation group of degree , then the permutation representation of is the linear representation of ρ : G → GL n ( K ) {\displaystyle \rho \colon G\to \operatorname {GL} _{n}(K)} which maps g ∈ G {\displaystyle g\in G} to the corresponding permutation matrix (here K {\displaystyle K} is an arbitrary field ). [ 2 ]
Frobenius group; Galois group of a polynomial; Jucys–Murphy element; Landau's function; Oligomorphic group; O'Nan–Scott theorem; Parker vector; Permutation group; Place-permutation action; Primitive permutation group; Rank 3 permutation group; Representation theory of the symmetric group; Schreier vector; Strong generating set; Symmetric ...
The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set.
A main problem in permutation codes is to determine the value of (,), where (,) is defined to be the maximum number of codewords in a permutation code of length and minimum distance . There has been little progress made for 4 ≤ d ≤ n − 1 {\displaystyle 4\leq d\leq n-1} , except for small lengths.
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