Question

Find the invariant factors of each of the groups in Exercise 5.

Short answer

$$\begin{gathered} \left(a\right)\left\{250\right\} \\ \left(b\right)\left\{2,6,36\right\} \\ \left(c\right)\left\{10,10,20,120\right\} \\ \left(d\right)\left\{2,60,60,300\right\} \end{gathered}$$

Step-by-step solution

Write the definition of invariant factors of G

If G is a finitely generated abelian group then,

$\mathrm{G}\cong {\mathrm{\mathbb{Z}}}^{\mathrm{r}}\oplus {\mathrm{\mathbb{Z}}}_{{\mathrm{m}}_1}\oplus .....\oplus {\mathrm{\mathbb{Z}}}_{{\mathrm{m}}_{\mathrm{s}}}$,

where for some integer r, we have,${\mathrm{m}}_1\ge {\mathrm{m}}_2....\ge {\mathrm{m}}_{\mathrm{s}}\ge 2$

Such that,$\mathrm{r}\ge 0$ and$\frac{{\mathrm{m}}_{\mathrm{i}+1}}{{\mathrm{m}}_{\mathrm{i}}}$

Thus, the integers$m_1,m_2,....,m_s$ are known as the invariant factors of the group.

Solution of part (a)

Consider ${\mathrm{\mathbb{Z}}}_{250}$.

Trivially invariant factor of${\mathrm{\mathbb{Z}}}_{250}$ is $\left\{250\right\}$.

Thus, the required result is $\left\{250\right\}$.

Solution of part (b)

Consider ${\mathrm{\mathbb{Z}}}_6+{\mathrm{\mathbb{Z}}}_{12}+{\mathrm{\mathbb{Z}}}_{18}$.

The trivially factor of${\mathrm{\mathbb{Z}}}_6+{\mathrm{\mathbb{Z}}}_{12}+{\mathrm{\mathbb{Z}}}_{18}$ is as follows:

$\begin{array}{rcl}{\mathrm{\mathbb{Z}}}_6+{\mathrm{\mathbb{Z}}}_{12}+{\mathrm{\mathbb{Z}}}_{18}& \cong & \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\right)\oplus \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2}\right)\\ & \cong & \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\right)\oplus \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\right)\\ & \cong & {\mathrm{\mathbb{Z}}}_6\oplus {\mathrm{\mathbb{Z}}}_6\oplus {\mathrm{\mathbb{Z}}}_{36}\end{array}$

Therefore, trivially invariant factors of${\mathrm{\mathbb{Z}}}_6\oplus {\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{18}$ are $\left\{2,6,36\right\}$.

Solution of part (c)

Consider ${\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}$.

The trivially factor of${\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}$ is as follows:

$\begin{array}{rcl}{\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}& \cong & \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_5\right)\\ & \cong & \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\\ & \cong & {\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{120}\end{array}$

Therefore, trivially invariant factors of${\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}$ are $\left\{10,10,20,120\right\}$.

Solution of part (d)

Consider ${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}$.

The trivially factor of${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}\oplus {\mathrm{\mathbb{Z}}}_{240}$ is as follows:

$\begin{array}{rcl}{\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}& \cong & \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\right)\oplus \left({\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{5^2}\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^4}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\\ & \cong & {\mathrm{\mathbb{Z}}}_2\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\oplus \left({{\mathrm{\mathbb{Z}}}_2}^4\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\right)\\ & \cong & {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{60}\oplus {\mathrm{\mathbb{Z}}}_{60}\oplus {\mathrm{\mathbb{Z}}}_{300}\end{array}$

Therefore, trivially invariant factors of${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}$ are ${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}$.