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[parent] $n$-divisible group (Definition)

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}\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_1,p_2,\ldots,p_k$.
Proof. If $ x$ is divisible by $ n$, write $ x=ny$, where $ y\in G$. Since $ p_i$ divides $ n$, write $ n=p_it_i$ where $ t_i$ is a positive integer. Then $ x=p_it_i(y)=p_i(t_iy)$. Since $ t_iy\in G$, $ x$ is divisible by $ p_i$. $ \qedsymbol$

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 $ pq$-divisible.
Proof. This follows from proposition 1 and the fact that if $ p\vert n$, $ q\vert n$ and $ \gcd(p,q)=1$, then $ pq\vert n$. $ \qedsymbol$
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\vert n$. Write $ n=p_1^{m_1}p_2^{m_2}\cdots p_k^{m_k}$. Then if $ G$ is $ p_i^{m_i}$-divisible for every $ i=1,\ldots, k$. Since $ p_i^{m_i}$ and $ p_j^{m_j}$ are coprime, $ G$ is $ n$-divisible by induction and proposition 3. $ \qedsymbol$

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



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Other names:  n-divisible group
Also defines:  n-divisible, $n$-divisible

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Cross-references: divisible group, induction, proposition, iff, coprime, decimal fractions, subset, group, divides, prime number, divisible, abelian group, integer, positive

This is version 2 of $n$-divisible group, born on 2007-08-08, modified 2007-08-13.
Object id is 9841, canonical name is NDivisibleGroup.
Accessed 1001 times total.

Classification:
AMS MSC: 20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous)
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