Linear Substitutions Commutative with a Given Substitution
Leonard Eugene Dickson · Proceedings of the London Mathematical Society · 1900
The object of this note is to determine the explicit form of all ra-ary linear homogeneous substitutions T with coefficients in the OF\_p n '] which are commutative with a particular one S.For the case n = 1, the number of such substitutions T has been determined by M. Jordan,* whose method of proof was, however, limited to the consideration of a particular example.By the use of convenient notations, we may treat the general case with equal ease and, moreover, avoid the separation of the proof into two successive stages.Following M. Jordan, I first give to S its canonical formf £,.. Let the characteristic determinant of $ bewhere F k (K), F t (K), ... are distinct polynomials belonging to, and irreducible* in, the OF [p"].We may exhibit the roots of F k (E) = 0 and of Ft (L) = 0 in the following notation:-To simplify the formulas, we suppose that F k and F t are the only irreducible factors of A (K).The method is, however, seen to be general.Corresponding to each partition of a and (3 into positive integers, * Traitd des Substitutions, pp.128-136.