Commutators in the Special and General Linear Groups

Citation

Thompson, Robert Charles (1960) Commutators in the Special and General Linear Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3BQM-3542. https://resolver.caltech.edu/CaltechETD:etd-07072006-084911

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let GL(n, K) denote the multiplicative group of all non-singular 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. 361-365; J. Math. Soc. of Japan, 3 (1951), p. 78-81). Theorem 2 supports the suggestion made by O. Ore (Proc. Amer. Math. Soc., 2 (1951), p. 307-314) that in a finite simple group, every element is a commutator.

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