Journal Article

ISOMORPHISM OF RELATIVE HOLOMORPHS AND MATRIX SIMILARITY

SZECHTMAN, FERNANDO, HERNANDEZ ALVARADO, ALBERTO J., GEBHARDT, VOLKER

2025

Overview

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary$\\alpha ,\\beta \\in \\mathrm {GL}(V)$, we consider the semidirect products$V\\rtimes \\langle \\alpha \\rangle $and$V\\rtimes \\langle \\beta \\rangle $, and show that if$V\\rtimes \\langle \\alpha \\rangle $and$V\\rtimes \\langle \\beta \\rangle $are isomorphic, then$\\alpha $must be similar to a power of$\\beta $that generates the same subgroup as$\\beta $; that is, if H and K are cyclic subgroups of$\\mathrm {GL}(V)$such that$V\\rtimes H\\cong V\\rtimes K$, then H and K must be conjugate subgroups of$\\mathrm {GL}(V)$. If we remove the cyclic condition, there exist examples of nonisomorphic , let alone nonconjugate, subgroups H and K of$\\mathrm {GL}(V)$such that$V\\rtimes H\\cong V\\rtimes K$. Even if we require that noncyclic subgroups H and K of$\\mathrm {GL}(V)$be abelian, we may still have$V\\rtimes H\\cong V\\rtimes K$with H and K nonconjugate in$\\mathrm {GL}(V)$, but in this case, H and K must at least be isomorphic. If we replace V by a free module U over${\\mathbb {Z}}/p^m{\\mathbb {Z}}$of finite rank, with$m>1$, it may happen that$U\\rtimes H\\cong U\\rtimes K$for nonconjugate cyclic subgroups of$\\mathrm {GL}(U)$. If we completely abandon our requirements on V , a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of$\\mathrm {Aut}(G)$such that$G\\rtimes H\\cong G\\rtimes K$. This criterion is satisfied by many groups.