Question

Prove that$\mathbf{\mathbb{Q}}$is an indecomposable group.

Short answer

Answer:

It has been proved that, $\mathrm{\mathbb{Q}}$is an indecomposable group.

Step-by-step solution

Definition of Indecomposable

A group $G$is said to be indecomposable if it is not the direct product of its two proper normal subgroups.

Prove that ℚ is indecomposable

Let $A_1,A_2$are non-zero subgroups of $\mathrm{\mathbb{Q}}$.

Let $a_i/b_i\in A_i$be non-zero elements.

Then, $a_1{a}_2=\left(a_2{b}_1\right)$.

Consider $a_1/b_1=\left(a_1{b}_2\right)·a_2/b_2$.

But this belongs to $A_1\cap A_2$, which implies $A_1,A_2$meet non-trivially.

So, $\mathrm{\mathbb{Q}}$cannot be the direct product of $A_1$and $A_2$.

Thus, it can be concluded that $\mathrm{\mathbb{Q}}$is indecomposable.