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'locally cyclic group'
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| Title of object: |
locally cyclic group |
| Canonical Name: |
GeneralizedCyclicGroup |
| Type: |
Definition |
| Created on: |
2003-07-23 11:06:47 |
| Modified on: |
2003-07-24 12:56:40 |
| Classification: |
msc:20Kxx |
| Synonyms: |
locally cyclic group=locally cyclic
locally cyclic group=generalized cyclic group
locally cyclic group=generalized cyclic
locally cyclic group=generalised cyclic
locally cyclic group=generalised cyclic group |
Revision comment (for changes between this and next version):
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Content:
\PMlinkescapeword{cyclic}
\PMlinkescapeword{subgroup}
\PMlinkescapeword{subgroups}
A {\em locally cyclic} (or {\em generalized cyclic}) group is a group in which any pair of elements generates a \PMlinkname{cyclic}{CyclicGroup} \PMlinkname{subgroup}{Subgroup}.
Every locally cyclic group is \PMlinkname{abelian}{AbelianGroup}.
If $G$ is a locally cyclic group, then every finite subset of $G$ generates a cyclic subgroup. Therefore, the only finitely-generated locally cyclic groups are the cyclic groups themselves. The group $(\mathbb{Q},+)$ is an example of a locally cyclic group that is not cyclic.
Subgroups and \PMlinkname{quotients}{QuotientGroup} of locally cyclic groups are also locally cyclic. |
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