abstract
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Minimal generating sets of groups, rings, and fields

Lorenz Halbeisen, Martin Hamilton, Pavel Ruzicka

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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 *Z*_{n} rings is computed.