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PRODUCTS OF SETS IN LINEAR GROUPS

PRODUCTS OF SETS IN LINEAR GROUPS DEMONSTRATIO MATHEMATICAVol. XXIINo 31989Jan AmbrosiewiczPRODUCTS OF SETS IN LINEAR GROUPSIn this paper we will generalize results containedin[2]. We will use the following notations : Z(G) denotes thecentrum of a group G, g denotes the conjugacy class of g inthe group G, I a ,...,a | (i=ln) denotes the det(a ).i1inijSL (n, K) denotes the set of matrices of GL(n, K) with determinant -1, A+B denotes the matrix= m>, V = diag(vw * w for i * j.JljInA 00 BK = {g € G :o(g) =mv ), W = diag(w , . . . , w ) with v * v ,InijThe remaining notation are standard.In the paper [1] it has been proved that K ^5 GL(n,K)for n £ 3 and for n = 2 if charK * 2. In this paper we willprove that if V,W e SL(n,K) or V,W € SL~(n,K) then SL(n,K) =(V W) 2 in the group GL(n,K) and if V,W e SL(n,K), where K isan algebraikyly closed field, then SL(n, K) = (V W) 2 in thegroup SL(n,K). We will prove also that SL(n,C) £(K2K2)2inthe group GL(n,C), where C is the field of complex numbers.We begin with the main theorem.T h e o r e http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

PRODUCTS OF SETS IN LINEAR GROUPS

Ambrosiewicz, Jan
Demonstratio Mathematica , Volume 22 (3): 18 – Jul 1, 1989
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Publisher
de Gruyter
Copyright
© by Jan Ambrosiewicz
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1989-0308
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXIINo 31989Jan AmbrosiewiczPRODUCTS OF SETS IN LINEAR GROUPSIn this paper we will generalize results containedin[2]. We will use the following notations : Z(G) denotes thecentrum of a group G, g denotes the conjugacy class of g inthe group G, I a ,...,a | (i=ln) denotes the det(a ).i1inijSL (n, K) denotes the set of matrices of GL(n, K) with determinant -1, A+B denotes the matrix= m>, V = diag(vw * w for i * j.JljInA 00 BK = {g € G :o(g) =mv ), W = diag(w , . . . , w ) with v * v ,InijThe remaining notation are standard.In the paper [1] it has been proved that K ^5 GL(n,K)for n £ 3 and for n = 2 if charK * 2. In this paper we willprove that if V,W e SL(n,K) or V,W € SL~(n,K) then SL(n,K) =(V W) 2 in the group GL(n,K) and if V,W e SL(n,K), where K isan algebraikyly closed field, then SL(n, K) = (V W) 2 in thegroup SL(n,K). We will prove also that SL(n,C) £(K2K2)2inthe group GL(n,C), where C is the field of complex numbers.We begin with the main theorem.T h e o r e

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 1989

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