# Tag Info

## New answers tagged abelian-groups

Accepted

### Is every ordered abelian group the additive group of an ordered ring?

No, and the classification of subgroups of $\mathbb{Q}$ can be used to find a counterexample. Consider the subgroup $A$ generated by $\frac{1}{p}$ as $p$ runs over all primes (or more generally ...
• 378k

### Exponent of noncyclic finite abelian group

Hint One of the $p_i$ has to be repeated in the expression for $G$ as a product of cyclic groups of order $p_i^{\alpha _i}$ (by the fundamental theorem of finite abelian groups). Otherwise we get ...
• 3,812
Accepted

### Does every embedding between finite free $\mathbb Z/p^2 \mathbb Z$-modules split?

This follows from the fact that $\mathbb{Z}/p^2\mathbb{Z}$ is injective as a module over itself. More generally, for any $n\neq 0$, $\mathbb{Z}/n\mathbb{Z}$ is injective over itself. To prove this, ...
• 302k

### Show that every commutative group of order 6 is cyclic

There are only two groups of order 6 less isomorphism (there are more groups, but the rest will be isomorphic to only two groups) $S_3$ and the $Z_6$. To reach this conclusion it is enough to employ ...

### Show that every commutative group of order 6 is cyclic

You can use Cauchy's theorem or its proof. By this theorem, there must be elements of order $2$ and elements of order $3$ in the group. Let $g_2$ and $g_3$ be elements of order $2$ and $3$, ...

### Show that every commutative group of order 6 is cyclic

The outline of a proof is like that: Assume there is no element of order 6. So all elements except e are of order 2 or 3. a. Show elements of order 3 exist in pairs. b. Show there is 4 element of ...
• 844
Accepted

### Do indecomposable abelian groups of inacessible cardinal rank exist?

According to Theorem 2.1 of the following paper, there are indecomposable torsion-free abelian groups of any infinite cardinality. Shelah, Saharon, Infinite Abelian groups, Whitehead problem and some ...
• 5,507

• 363k

• 9,908

• 216
Accepted

### What are the torsion coefficients of $\Bbb Z_{30}\oplus \Bbb Z_{18}\oplus\Bbb Z_{75}?$

Using only what you know, we have $$(30,18,75) \to (2,3,5,2,9,3,25) \to (3,30,450).$$.
• 81.1k
Lemma: Let $G$ be a group and $H\le G, K\triangleleft G$ then $H\vee K=HK$ Proof: $HK\subset H\vee K$. $HK\le G$. $H\vee K$ is the smallest subgroup of $G$ containing both $H$ and $K$. $\square$ ...
One way to define the (internal) direct product $G=H\bowtie K$ of two subgroups $H,K$ of a group is that all of the following is satisfied: $G=HK$. $H\cap K=\{e\}.$ $H,K\unlhd G$. It is a standard ...