Question

Show by example that a homomorphic image of an indecomposable group need not be indecomposable.

Short answer

Answer:

It has been proved that, a homomorphic image of an indecomposable group need not be indecomposable.

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.

Consider two homomorphic groups

Consider the group $\mathrm{\mathbb{Z}}$.

Define a function $f:\mathrm{\mathbb{Z}}\to {\mathrm{\mathbb{Z}}}_6$such that, $f\left(k\right)=kmod6$.

Then, ${\mathrm{\mathbb{Z}}}_6$is a homomorphic image of $\mathrm{\mathbb{Z}}$.

Prove that ℤ is indecomposable

Any two non-zero subgroups of $\mathrm{\mathbb{Z}}$are cyclic.

So, the two subgroups coincide with$<m>$and $<n>$for some $m,n\in \mathrm{\mathbb{Z}}$.

Hence, their intersection is non-trivial say $<k>$, where $k=lcm(m,n)$.

Therefore, $\mathrm{\mathbb{Z}}$is not the direct product of these subgroups.

Thus, $\mathrm{\mathbb{Z}}$is indecomposable.

Prove that ℤ6 is decomposable

${\mathrm{\mathbb{Z}}}_6$is direct product of two of its subgroups ${\mathrm{\mathbb{Z}}}_2$and ${\mathrm{\mathbb{Z}}}_3$.

Therefore, ${\mathrm{\mathbb{Z}}}_6$is decomposable

Thus, it can be concluded that a homomorphic image of an indecomposable group need not be indecomposable.