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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.
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]
This permutation can be described by starting with the tiling of the Klein quartic by 56 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of those edges of the original ...
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. This is because for a non-transitive action, either the orbits of G form a nontrivial partition preserved by G , or the group action is trivial, in which case all nontrivial partitions ...
Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the positions of the zeroes in the Lehmer code are the positions of the right-to-left minima (in the example positions the 4, 8, 9 of the values 1, 2, 5); this allows computing the distribution ...
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 ...
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