Some Facts
- Every cyclic group is locally cyclic, and every locally cyclic group is abelian.
- Every finitely-generated locally cyclic group is cyclic.
- Every subgroup and quotient group of a locally cyclic group is locally cyclic.
- A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
- A group is locally cyclic if and only if its lattice of subgroups is distributive (Ore 1938).
- The torsion-free rank of a locally cyclic group is 0 or 1.
Read more about this topic: Locally Cyclic Group
Famous quotes containing the word facts:
“In the fevered state of our country, no good can ever result from any attempt to set one of these fiery zealots to rights, either in fact or principle. They are determined as to the facts they will believe, and the opinions on which they will act. Get by them, therefore, as you would by an angry bull; it is not for a man of sense to dispute the road with such an animal.”
—Thomas Jefferson (1743–1826)
“One would appear ridiculous who would say, that it is only probable the sun will rise to-morrow, or that all men must die; though it is plain we have no further assurance of these facts than what experience affords us.”
—David Hume (1711–1776)
Related Phrases
Related Words