Similarity of matrices over finite rings

Abstract

It is shown that questions of similarity of certain invertible matrices over a finite ring can be reduced to questions of similarity over finite fields through the application of canonical epimorphisms. Suprunenko has shown in [3] that two invertible matrices over ZIZm whose orders are relatively prime to m are similar if and only if their canonical images are similar over the fields Z/Zp for each prime divisor p of m. An analogous result holds for invertible matrices over any finite commutative ring with identity. Preliminaries. If R is a finite commutative ring with identity, then R is uniquely a ring direct product of finite local rings [1, Theorem 8.7, p. 90]. Suppose that R=]7Lf Ri, where Ri is a finite local ring with maximal ideal Mi. Each Ri has cardinality pzi for some prime p and has associated with it a canonical projection, hi: Ri -RiMi = GF(p ). Setting ki=GF(pfi) we will say that the finite fields {ki:i=l, 2, , t} are the fields associated with R. Observe that the decomposition of R carries over to the general linear group of degree n over R yielding GLn(R)FJ7LJ1GL,(RJ). Furthermore, for each i, the projection hi induces an epimorphism, hi: GL (Ri) -GLn(ki). If GLn(R,) is taken as the group of n by n invertible matrices over Ri, then hi is simply reduction modulo Mi. Note that the kernel of hi, Ki, has cardinality a power of pi and thus is a solvable group. The following corollary to P. Hall's extension of the Sylow theorems [2, Theorem 9.3.1, p. 141] is the key result needed for Theorems 1 and 2. Observation. Let G be a finite group with solvable normal subgroup K and let C=G/K={glg E G}. Let o and f, be elements of G with (IoI, IkI)= 1=(1II, IKI). Then o-j implies oc-z . Received by the editors April 12, 1972. AMS (MOS) subject classifications (1970). Primary 13H99, 15A33, 15A21, 20D20, 20H25.