Recognition of finite groups by a set of orders of their elements

We consider a finite matrix group N ⊂ GL 8 (R) with 3 4 · 2 16 elements, which is a subgroup of the infinite group Q,C ⊂ GL 8 (C), where Q is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group Q,C similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically.We study the structure of the group N and use this group to construct spherical orbit codes on the unit Euclidean sphere in R 8 . These codes have code distance less than 1. One of them has 3 2 · 2 8 = 2304 elements and its squared Euclidean code distance is 0.293.

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“…Hence the order of the intersection of N with the base of the wreath product divides 2 8 . This implies that |N| divides 2 8 |S 8 |, i.e.…”

Section: The Structure Of the Group N (A Priori Knowledge)mentioning

confidence: 97%

Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

et al. 2005

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“…In [1][2][3][4], the recognition problem for simple groups U 3 (q) was solved for all even q and for q equal to 3,5,7,9,11. The main goal of this paper is to argue for the validity of the following: THEOREM 1.…”

Section: Introductionmentioning

confidence: 98%

Recognition of simple groups U 3(Q) by element orders

Zavarnitsin

2006

Algebr Logic

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“…The recognizability problem for L 3 (q) and U 3 (q) was investigated in [2][3][4][5][6][7][8][9] and was ultimately settled in [10,11]. Whether groups L n (2) are recognizable by spectrum for various particular values of n was explored in [12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%