subgoups of locally cyclic groups are locally cyclic

Theorem 1.

A group G is locally cyclic iff every subgroupMathworldPlanetmathPlanetmath HG is locally cyclic.


Let G be a locally cyclic group and H a subgroup of G. Let S be a finite subset of H. Then the group S generated by S is a cyclic subgroup of G, by assumption. Since every element a of S is a product of elements or inversesMathworldPlanetmathPlanetmathPlanetmath of elements of S, and S is a subset of group H, aH. Hence S is a cyclic subgroup of H, so H is locally cyclic.

Conversely, suppose for every subgroup of G is locally cyclic. Let H be a subgroup generated by a finite subset of G. Since H is locally cyclic, and H itself is finitely generatedMathworldPlanetmathPlanetmathPlanetmath, H is cyclic, and therefore G is locally cyclic. ∎

Title subgoups of locally cyclic groups are locally cyclic
Canonical name SubgoupsOfLocallyCyclicGroupsAreLocallyCyclic
Date of creation 2013-03-22 17:14:46
Last modified on 2013-03-22 17:14:46
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Theorem
Classification msc 20E25
Classification msc 20K99