Linear semigroups of polynomial growth in positive characteristic
type: Item
instance of: scholarly article
title: Linear semigroups of polynomial growth in positive characteristic
zbMATH Open document ID: 0854.20072
DOI: 10.1016/0022-4049(95)00067-4
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-20T00:00:00Z
review text: The problem of polynomial growth for linear semigroups (over fields) attracted attention of different mathematicians. In the present paper this problem is solved completely for any finitely generated linear semigroup over a field of positive characteristic. Let \(K\) be a field, \(\text{ch }K>0\), and let \(M_n(k)\) be the full matrix semigroup over \(K\). The main result of the paper is the following. Suppose \(S\) is a finitely generated subsemigroup of \(M_n(K)\). Then \(S\) has polynomial growth if and only if every nonempty intersection \(S\cap D\) with a maximal subgroup \(D\) of \(M_n(K)\) has a finitely generated nilpotent-by-finite group of fractions. Moreover, the degree of growth of such \(S\) is bounded by a function \(f(n,r)\) of \(n\) and of the maximal rank \(r\) of a free commutative subsemigroup of \(S\). For \(K\) an arbitrary field, the author published analogous (but not so complete) results [in J. Aust. Math. Soc., Ser. A 60, No. 1, 18-30 (1996; Zbl 0852.20055)].
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 908492
zbMATH Keywords: polynomial growth
zbMATH Keywords: linear semigroups
zbMATH Keywords: matrix semigroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: finitely generated nilpotent-by-finite group of fractions
zbMATH Keywords: degree of growth
zbMATH Keywords: free commutative subsemigroups
zbMATH Keywords: finitely generated subsemigroups
author: Jan Okniński
reviewed by: Josif S. Ponizovskii
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00067-4
OpenAlex ID: W1992014621
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description: scientific article; zbMATH DE number 908492