Recognizing cyclic matrices and a conjecture of J.G. Thompson
Abstract
In 2006 J.G. Thompson conjectured: "If F is a field and A is in GL(n,F), then there is a permutation matrix P such that AP is cyclic, that is, the minimal polynomial of AP is also its characteristic polynomial" (open problem 16.95 in the Kourovka Notebook). The present note provides a simple criterion for a matrix to be cyclic and uses this to prove Thompson's conjecture. ERRATA I am indebted to Alexander Stasinski (Durham University) for the following observations. Suppose n > 2 and J is the n x n all 1's matrix over a field of characteristic not 2. Then A := J - I has the minimal polynomial (X + 1)(X - n + 1). Thus A is invertible and not cyclic even though A satisfies condition (iv) of the Proposition. The error lies in the claim towards the end of the proof that "these particular row and column errors do not change the determinants ...".
Keywords
Cite
@article{arxiv.1606.02238,
title = {Recognizing cyclic matrices and a conjecture of J.G. Thompson},
author = {John D. Dixon},
journal= {arXiv preprint arXiv:1606.02238},
year = {2017}
}
Comments
I am withdrawing this paper since the implication (iv) => (i) in the Proposition is false for n > 2. As a consequence the conjecture of J.G. Thompson remains open