Definition

A locally cyclic group is a group in which every finite subset generates a cyclic subgroup.

Properties

From the definition we see that every finitely generated locally cyclic group (and, in particular, every finite locally cyclic group) is cyclic.

The following can all be shown to be equivalent for a group :

is locally cyclic.
For all $a,b\in G$ , the subgroup ${\left\langle a,b\right\rangle}$ is cyclic.
is the union of a chain of cyclic subgroups.
The lattice of subgroups of is distributive.
embeds in $\mathbb{Q}$ or $\mathbb{Q}/\mathbb{Z}$ .
is isomorphic to a subgroup of a quotient of $\mathbb{Q}$ .
is involved in $\mathbb{Q}$ .

From the last of these equivalent properties it is clear that every locally cyclic group is countable and abelian, and that subgroups and quotients of locally cyclic groups are locally cyclic.

"locally cyclic group" is owned by yark.

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See Also: cyclic group, abelian group, group, locally $\cal P$

Other names:

locally cyclic, generalized cyclic group, generalized cyclic, generalised cyclic, generalised cyclic group

Attachments:: subgoups of locally cyclic groups are locally cyclic (Theorem) by rspuzio

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Cross-references: abelian, countable, isomorphic, lattice of subgroups, chain, union, cyclic, finitely generated, cyclic subgroup, subset, finite, group
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This is version 21 of locally cyclic group, born on 2003-07-23, modified 2007-06-13.
Object id is 4497, canonical name is GeneralizedCyclicGroup.
Accessed 6663 times total.

Classification:

AMS MSC:	20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous)
	20E25 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Local properties)

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