Difference between revisions of "Locally cyclic group"
From Encyclopedia of Mathematics
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| − | * Marshall Hall, ''The Theory of Groups'', reprinted American Mathematical Society (1976) ISBN 0-8218-1967-4 | + | * Marshall Hall jr, ''The Theory of Groups'', reprinted American Mathematical Society (1976)[1959] {{ISBN|0-8218-1967-4}} {{ZBL|0084.02202}} {{ZBL|0354.20001}} |
Latest revision as of 20:32, 18 November 2023
2020 Mathematics Subject Classification: Primary: 20E [MSN][ZBL]
A group in which every finitely generated subgroup is cyclic. In such a group, either every element is of finite order (periodic), or no element other than the identity is (aperiodic). The additive group of rational numbers $\mathbb{Q}^+$ is an aperiodic example, and the group $\mathbb{Q}/\mathbb{Z}$ is a periodic example. The lattice of subgroups of a group is a distributive lattice if and only if the group is locally cyclic.
References
- Marshall Hall jr, The Theory of Groups, reprinted American Mathematical Society (1976)[1959] ISBN 0-8218-1967-4 Zbl 0084.02202 Zbl 0354.20001
How to Cite This Entry:
Locally cyclic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_cyclic_group&oldid=35797
Locally cyclic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_cyclic_group&oldid=35797