### ON THE SETS CVCV IN THE GROUP SL(n, K)

Jan Ambrosiewicz
1995 Demonstratio Mathematica
In  has been proved that (1) if A € SL(n, K) is not a scalar and K has at least four elements, then A is a commutator of SL(n,K). In a present paper we shall prove that (2) if A € SL(n,K) is not a scalar and n < \K\ -1 or char/i = 0, then there exists class CV such that A G CyCy-x, where Cy denotes the conjugacy class of V € SL{n, K). Observe that the condition A 6 CyCy-1 implies that A is a commutator. Hence if n < \K\ -1 or char K = 0, then the second theorem is stronger than the first
more » ... than the first one. We will give a condition for which there exists V € SL(n, K) such that SL(n, K) = CyCv-That is an answer to the question: whether there exists Cv such that CVCV = SL(n,K), (see  . p. 66). Simultaneously we give a new proof of the equality PSL(n, K) = CyCy, different from that in . The notation is standard. In addition we shall denote by SLi(n,K) the subgroup of matrices of determinant equal to ± 1. We shall use the following theorem.