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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.
The Mathieu group M 24 has a 5-fold transitive permutation representation on 24 points. The corresponding linear representation over the complex numbers is the sum of the trivial representation and a 23-dimensional irreducible representation.
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.
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 ...
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 ]
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In the grid, the sum to the right of the equal sign shows the lengths of the three orbits of the stabilizer of a point of the permutation group. For example, the expression 15 = 1+6+8 in the first row of the table under the heading means that the permutation group for the first row has degree 15, and the lengths of three orbits of the ...