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Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
All 11, onto functions f : S → S (permutations of S), where S is any non‐empty set, form a group under composition of functions. See Section 2.5.3. In particular, if S = {1, 2, 3, . . . , n}, the group of permutations of S is called the symmetric group, Sn. In Sn, each permutation can be written as a product of cycles. A cycle is a permutation σ = (i1i2...ik) $ (i_{1} i_{2} . . . i_{k} ) $ , where σ(i1) = i2, σ(i2) = i3, . . . , σ(ik) = i1. Each cycle of length greater than 1 can be written as a product of transpositions (cycles of length 2). A permutation is even (odd) if it can be written as the product of an even (odd) number of transpositions. (Every permutation is either even or odd.) The set of all even permutations in Sn is a normal subgroup, An, of Sn. The group An is called the alternating group on n elements.
Mathematical Morphology with Noncommutative Symmetry Groups
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
The group of the tetrahedron is A4that of the cube and the octahedron is S4, and that of the dodecahedron and icosahedron isA5. Here An denotes the alternating group on n points, that is, the subgroup of Sn containing only even permutations [2].
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
(The determinant (call it D) on the right is known as the Vander Monde determinant. If we set, for i < j, ai = aj in D, then two columns become identical, and hence D = 0. This means that (ai − aj) is a factor of D. Further the coefficient of a2a32…ann−1 on both sides is 1. Hence the last equality.) The even permutations in Sn form a subgroup An of Sn, called the alternating group of degree n.Proof. If σ1 and σ2 are even permutations in Sn, then each of them is a product of an even number of transpositions and hence so is their product. Further if σ=t1t2…tr, where each ti is a transposition, then σ−1=tr−1tr−1−1…t2−1t1−1. Hence the inverse of an even permutation is even. Further the identity permutation is even since for any transposition t, e = t° t = t2.A permutation is odd or even according to whether it is expressible as a product of an odd number or even number of transpositions.
Diagnosability of arrangement graphs with missing edges under the MM* model
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
Let , and let be the symmetric group on containing all permutations of . The alternating group is the subgroup of containing all even permutations. It is well known that is a generating set for . The n-dimensional alternating group graph is the graph with vertex set = in which two vertices u, v are adjacent if and only if or , .