Showing papers on "Hyperoctahedral group published in 1983"
TL;DR: It is proved that the group of permutations generated by the DES-like functions is exactly the alternating group of the set of binary n vectors.
Abstract: A set of transformations on binary vectors of length n is defined. These transformations are similar to those of the data encryption standard (DES) and therefore are called DES-like functions. It is proved that the group of permutations generated by the DES-like functions is exactly the alternating group of the set of binary n vectors.
45 citations
TL;DR: In this article, the relations between the irreducible representations of the hyperoctahedral group in four dimensions and the low-dimensional representation of the orthogonal groups O(4) and SO(4).
Abstract: In this paper we describe the relations between the irreducible representations of the hyperoctahedral group in four dimensions and irreducible, low‐dimensional representations of the orthogonal groups O(4) and SO(4).
17 citations
TL;DR: In this article, a simple symmetric set of non-isotropic lines is shown to be a simple automorphism group of a simple set of symmetric sets of vectors.
Abstract: Orthogonal groups are considered as automorphism groups of some symmetric sets of vectors. From this point of view, we can prove the well-known theorem of simplicity on orthogonal groups. (The cases for the other classical groups are given in [5].) The proof consists of two steps. The first step which will be given in 1 is to show that a transitive symmetric set of non-isotropic lines (of a certain type) is simple. After a short review on simple symmetric set is given, we will show the above fact. A point here is that it is so when dim V is 3. The second step is to show that the group of displacements of the simple symmetric set is a simple group, which will be given in 2. A useful supplement to the main theorem on simple symmetric sets will be found, and using it we can show the above fact when dim V>5.
5 citations
4 citations
IBM1
TL;DR: The APL function REVERSE and a certain defined function generate a group of order four on permutation vectors, known as D4, the fourth dihedral group, and the resulting isomorphism maps GRADE-DOWN and its inverse to the ninety degree rotations.
Abstract: The APL function REVERSE and a certain defined function generate a group of order four on permutation vectors. The group is extended with the GRADE-UP function to produce GRADE-DOWN and the order eight group known as D4, the fourth dihedral group. The symmetries of a square are generated by REVERSE and TRANSPOSE on square matrices. The resulting isomorphism maps GRADE-DOWN and its inverse to the ninety degree rotations. APL's powerful formalism easily reveals the isomorphisms between the domains of algebra and geometry.
2 citations