Equivalence of definitions of locally cyclic periodic group
This article gives a proof/explanation of the equivalence of multiple definitions for the term locally cyclic periodic group
View a complete list of pages giving proofs of equivalence of definitions
The following are equivalent for a group:
- It is a Locally cyclic group (?) as well as a Periodic group (?): every element has finite order.
- It is isomorphic to a restricted direct product of groups, where for each prime , there is either one cyclic group of order a power of appearing or one p-Quasicyclic group (?) appearing.
Given A periodic locally cyclic group .
To prove: is isomorphic to a restricted direct product of groups, where for each prime , there is either one cyclic group of order a power of appearing or one p-quasicyclic group appearing.
Proof: For each prime , let be the subgroup of comprising elements whose order is a power of . Note that this is a subgroup, because the conditions for products, inverses, and identity element are satisfied.
- is a restricted direct product of the s: Since every element of has finite order, it generates a finite cyclic group which is a direct product of cyclic groups of prime power order. In particular, the element itself is a product of commuting elements of prime power order. Thus, the s generate . Also, each intersects trivially the subgroup generated by all the others, because the order of an element in is a power of and the order of any element generated by the other pieces is relatively prime to .
- Each is locally cyclic and periodic: This follows because both conditions inherit to subgroups.
- If finite, is cyclic: This follows immediately from the previous step, because a finite locally cyclic group is cyclic.
- If infinite, is the -quasicyclic group, i.e., it is isomorphic to the group of all -powered complex roots of unity under multiplication: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]