Equivalence of definitions of locally cyclic periodic group

From Groupprops
Jump to: navigation, search
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

Statement

The following are equivalent for a group:

  1. It is a Locally cyclic group (?) as well as a Periodic group (?): every element has finite order.
  2. It is isomorphic to a restricted direct product of groups, where for each prime p, there is either one cyclic group of order a power of p appearing or one p-Quasicyclic group (?) appearing.

Proof

Given A periodic locally cyclic group G.

To prove: G is isomorphic to a restricted direct product of groups, where for each prime p, there is either one cyclic group of order a power of p appearing or one p-quasicyclic group appearing.

Proof: For each prime p, let G_p be the subgroup of G comprising elements whose order is a power of p. Note that this is a subgroup, because the conditions for products, inverses, and identity element are satisfied.

  1. G is a restricted direct product of the G_ps: Since every element of G 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 G_ps generate G. Also, each G_p intersects trivially the subgroup generated by all the others, because the order of an element in G_p is a power of p and the order of any element generated by the other pieces is relatively prime to p.
  2. Each G_p is locally cyclic and periodic: This follows because both conditions inherit to subgroups.
  3. If finite, G_p is cyclic: This follows immediately from the previous step, because a finite locally cyclic group is cyclic.
  4. If infinite, G_p is the p-quasicyclic group, i.e., it is isomorphic to the group of all p^{th}-powered complex roots of unity under multiplication: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]