# $n$-divisible group

Let $n$ be a positive integer and $G$ an abelian group^{}. An element $x\in G$ is said to be divisible by $n$ if there is $y\in G$ such that $x=ny$.

By the unique factorization of $\mathbb{Z}$, write $n={p}_{1}^{{m}_{1}}{p}_{2}^{{m}_{2}}\mathrm{\cdots}{p}_{k}^{{m}_{k}}$ where each ${p}_{i}$ is a prime number^{} (distinct from one another) and ${m}_{i}$ a positive integer.

###### Proposition 1.

If $x$ is divisible by $n$, then $x$ is divisible by ${p}_{\mathrm{1}}\mathrm{,}{p}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{p}_{k}$.

###### Proof.

If $x$ is divisible by $n$, write $x=ny$, where $y\in G$. Since ${p}_{i}$ divides $n$, write $n={p}_{i}{t}_{i}$ where ${t}_{i}$ is a positive integer. Then $x={p}_{i}{t}_{i}(y)={p}_{i}({t}_{i}y)$. Since ${t}_{i}y\in G$, $x$ is divisible by ${p}_{i}$. ∎

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 $D\subseteq \mathbb{Q}$ 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 ${n}^{s}$-divisible for every non-negative integer $s$.

###### Proposition 3.

Suppose $p$ and $q$ are coprime^{}, then $G$ is $p$-divisible and $q$-divisible iff it is $p\mathit{}q$-divisible.

###### Proof.

This follows from proposition^{} 1 and the fact that if $p|n$, $q|n$ and $\mathrm{gcd}(p,q)=1$, then $pq|n$. ∎

###### Proposition 4.

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

###### Proof.

Suppose $G$ is $n$-divisible. By proposition 1, every element $x\in G$ is divisible by $p$, so that $G$ is $p$-divisible. Conversely, suppose $G$ is $p$-divisible for every $p|n$. Write $n={p}_{1}^{{m}_{1}}{p}_{2}^{{m}_{2}}\mathrm{\cdots}{p}_{k}^{{m}_{k}}$. Then if $G$ is ${p}_{i}^{{m}_{i}}$-divisible for every $i=1,\mathrm{\dots},k$. Since ${p}_{i}^{{m}_{i}}$ and ${p}_{j}^{{m}_{j}}$ are coprime, $G$ is $n$-divisible by induction^{} 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 |