Minimal generating sets of groups, rings, and fields

  • Lorenz Halbeisen Institut f&#252r Informatik und angewandte Mathematik, Universit¨at Bern, Neubr&#252ckstrasse 10, CH-3012 Bern, Switzerland
  • Martin Hamilton Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
  • Pavel Ruzicka Katedra Algebry, Univerzita Karlova v Praze, Sokolovsk&#224 83, 186 75 Praha 8, Czech Republic
Keywords: Minimal generating sets, cyclic groups, finite rings, field extensions

Abstract

A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup (or subring, or subfield) containing X is the group (ring, field) itself. A generating set X is called minimal generating, if X does not properly contain any generating set. The existence and cardinalities of minimal generating sets of various groups, rings, and fields are investigated. In particular it is shown that there are groups, rings, and fields which do not have a minimal generating set. Among other result, the cardinality of minimal generating sets of finite abelian groups and of finite products of Zn rings is computed.

Quaestiones Mathematicae 30(2007), 355–363
Published
2007-10-10
Section
Articles

Journal Identifiers


eISSN: 1727-933X
print ISSN: 1607-3606