Powers of 1-cyclic conjugacy classes in the groups GLn(F) and SLn(F) Original Research Article

Let A set membership, variant GLn(F), where F is a field. We say that A is 1-cyclic if A is similar to a matrix of the form A′ = diag{A1, A2, …, Ak}, where Ai set membership, variant GLli(F) is cyclic for 1 less-than-or-equals, slant i less-than-or-equals, slant k, l1 set membership, variant {0, 1}, and li greater-or-equal, slanted 2 for 2 less-than-or-equals, slant i less-than-or-equals, slant k. It is shown that if A set membership, variant GLn(F) is 1-cyclic, where n greater-or-equal, slanted 2 and F greater-or-equal, slanted 4, then every nonscalar matrix M set membership, variant GLn(F) whose determinant equals (det A)4 is the product of four matrices which are similar to A under matrices of SLn(F). The problem of expressing a scalar matrix as a product of similar 1-cyclic matrices is also discussed. The above result is applied to problems of factorizing matrices in the group SLn(F) into products of unipotent matrices of index 2, and into products of matrices of (fixed) finite order.