Quaestiones Mathematicae

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Minimal generating sets of groups, rings, and fields

Lorenz Halbeisen, Martin Hamilton, Pavel Ruzicka


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
AJOL African Journals Online