Products of conjugacy classes of two by two matrices
type: Item
instance of: scholarly article
title: Products of conjugacy classes of two by two matrices
zbMATH Open document ID: 0840.20040
DOI: 10.1016/0024-3795(94)00005-X
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 1995-12-18T00:00:00Z
review text: The covering number \(\text{cn} (G)\) and the extended covering number \(\text{ecn} (G)\) of a simple noncommutative group is the least \(k\) such that \(\text{cn} (G)\leq k\) if \(C^k= G\) for every nontrivial conjugacy class \(C\) in \(G\), and \(\text{ecn} (G)\leq k\) if \(C_1\cdots C_k= G\) for any nontrivial conjugacy classes \(C_j\) in \(G\), respectively. The authors extend these notions to arbitrary groups: Let \(C_1, \dots, C_k\) be any conjugacy classes in \(G\) such that the normal subgroup generated by each of them contains \([G,G ]\). Then the extended covering number \(\text{ecn} (G)\) is the least \(k\) such that the product \(C_1\cdots C_k\) contains every similarity class \(C_0\) with \(C_0= C_1 \cdots C_k \bmod [G,G ]\), and the covering number \(\text{cn} (G)\) is the least \(k\) satisfying this condition for equal classes \(C_1= \dots= C_k\). Clearly, \(\text{cn} (G)\leq \text{ecn} (G)\). The authors compute the covering number and the extended covering number for the groups \(G= \text{PSL}_2 (F)\), \(\text{GL}_2 (F)\), \(\text{PGL}_2 (F)\), and \(\text{SL}_2 (F)\) over an arbitrary field \(F\). It turns out, that \(\text{cn} (G)\) and \(\text{ecn} (G)\) depend on certain properties of the field. So \(\text{cn} (G)\) and \(\text{ecn} (G)\) may be 2, 3, 4 or 5 depending on whether the field is quadratically closed, a \(C_1 (2)\)-field, or not a \(C_1 (2)\)-field.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 826714
zbMATH Keywords: covering number
zbMATH Keywords: extended covering number
zbMATH Keywords: conjugacy classes
zbMATH Keywords: similarity class
reviewed by: Huberta Lausch
MaRDI profile type: Publication
cites work: Products of conjugacy classes in groups
cites work: Q4771141
cites work: On a question of M. Newman on the number of commutators
cites work: Q5675788
cites work: Products of cyclic conjugacy classes in the groups \(\text{PSL}(n,F)\)
cites work: Q4010272
cites work: Q3485004
description: scientific article; zbMATH DE number 826714