subgoups of locally cyclic groups are locally cyclic
Let be a locally cyclic group and a subgroup of . Let be a finite subset of . Then the group generated by is a cyclic subgroup of , by assumption. Since every element of is a product of elements or inverses of elements of , and is a subset of group , . Hence is a cyclic subgroup of , so is locally cyclic.
Conversely, suppose for every subgroup of is locally cyclic. Let be a subgroup generated by a finite subset of . Since is locally cyclic, and itself is finitely generated, is cyclic, and therefore is locally cyclic. ∎
|Title||subgoups of locally cyclic groups are locally cyclic|
|Date of creation||2013-03-22 17:14:46|
|Last modified on||2013-03-22 17:14:46|
|Last modified by||rspuzio (6075)|