n-divisible group

Let n be a positive integer and G an abelian groupMathworldPlanetmath. An element xG is said to be divisible by n if there is yG such that x=ny.

By the unique factorization of , write n=p1m1p2m2pkmk where each pi is a prime numberMathworldPlanetmath (distinct from one another) and mi a positive integer.

Proposition 1.

If x is divisible by n, then x is divisible by p1,p2,,pk.


If x is divisible by n, write x=ny, where yG. Since pi divides n, write n=piti where ti is a positive integer. Then x=piti(y)=pi(tiy). Since tiyG, x is divisible by pi. ∎

Definition. An abelian group G such that every element is divisible by n is called an n-divisible group. Clearly, every group is 1-divisible.

For example, the subset D of all decimal fractions is 10-divisible. D is also 2 and 5-divisible. In general, we have the following:

Proposition 2.

If G is n-divisible, it is also ns-divisible for every non-negative integer s.

Proposition 3.

Suppose p and q are coprimeMathworldPlanetmath, then G is p-divisible and q-divisible iff it is pq-divisible.


This follows from propositionPlanetmathPlanetmathPlanetmath 1 and the fact that if p|n, q|n and gcd(p,q)=1, then pq|n. ∎

Proposition 4.

G is n-divisible iff G is p-divisible for every prime p dividing n.


Suppose G is n-divisible. By proposition 1, every element xG is divisible by p, so that G is p-divisible. Conversely, suppose G is p-divisible for every p|n. Write n=p1m1p2m2pkmk. Then if G is pimi-divisible for every i=1,,k. Since pimi and pjmj are coprime, G is n-divisible by inductionMathworldPlanetmath and proposition 3. ∎

Remark. G is a divisible group iff G is p-divisible for every prime p.

Title n-divisible group
Canonical name NdivisibleGroup
Date of creation 2013-03-22 17:27:30
Last modified on 2013-03-22 17:27:30
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 20K99
Synonym n-divisible group
Defines n-divisible
Defines n-divisible