Citation
Thompson, Robert Charles (1960) Commutators in the special and general linear groups. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd07072006084911
Abstract
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Let GL(n, K) denote the multiplicative group of all nonsingular nxn matrices with coefficients in a field K; SL(n, K) the subgroup of GL(n, K) consisting of all matrices with determinant unity; C(n, K) the centre of SL(n, K); PSL(n, K) the factor group SL(n, K)/C(n, K); I(n) the nxn identity matrix; GF(pn) the finite field with pn elements. We determine when every element of SL(n, K) is a commutator of SL(n, K) or of GL(n, K). Theorem 1. Let A [...] SL(n, K). Then it follows that A is a commutator [...] of SL(n, K) unless: (i) n = 2 and K = GF(2); (ii) n = 2 and K = GF(3); or (iii) K has characteristic zero and A = [...] where a is a primitive nth root of unity in K and n [...] 2 (mod 4). In case (i), SL(2, GF(2)) properly contains its commutator subgroup. In case (ii), SL(2, GF(3)) properly contains its commutator subgroup. Furthermore, every element of SL(2, GF(3)) is a commutator of GL(2, GF(3)). In case (iii), [...] is always a commutator of GL(n, K). Moreover, aIn is a commutator of SL(n, K) when, and only when, the equation 1 = x2 + y2 has a solution x, y [...] K. Hence: Theorem 2. Whenever PSL(n, K) is simple, every element of PSL(n, K) is a commutator of PSL(n, K). Theorem 1 simplifies and extends results due to K. Shoda (Jap. J. Math., 13 (1936), p. 361365; J. Math. Soc. of Japan, 3 (1951), p. 7881). Theorem 2 supports the suggestion made by O. Ore (Proc. Amer. Math. Soc., 2 (1951), p. 307314) that in a finite simple group, every element is a commutator.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1960 
Record Number:  CaltechETD:etd07072006084911 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd07072006084911 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2820 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  26 Jul 2006 
Last Modified:  26 Dec 2012 02:54 
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