Chapter 9: Question: (page 297)

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

Short Answer

Expert verified

${250}$
${2, 6, 36}$
${10, 10, 20, 120}$
${2, 60, 60, 300}$

Step by step solution

Write the definition of invariant factors of G

If $G$ is a finitely generated abelian group then,

$G ≅ ϕ^{r} \oplus ϕ_{m_{1}} \oplus \dots \oplus ϕ_{m_{s}}$ ,

where for some integer $r$ , we have,

$m_{1} \geq m_{2} \dots \geq m_{s} \geq 2$

Such that, $r \geq 0$ and $\frac{m_{i + 1}}{m_{i}}$

Thus, the integers $m_{1}, m_{2}, \dots, m_{s}$ are known as the invariant factors of the group.

Solution of part (a)

Consider $ϕ_{250}$

Trivially invariant factor of $ϕ_{250}$ is ${250}$ .

Thus, the required result is ${250}$ .

Solution of part (b)

Consider $ϕ_{6} \oplus ϕ_{12} \oplus ϕ_{18}$

The trivially factor of $ϕ_{6} \oplus ϕ_{12} \oplus ϕ_{18}$ is as follows:

$ϕ_{6} \oplus ϕ_{12} \oplus ϕ_{18} ≅ (ϕ_{2} \oplus ϕ_{3}) \oplus (ϕ_{2^{2}} \oplus ϕ_{3}) \oplus (ϕ_{2} \oplus ϕ_{3^{2}})$

$≅ ({\dot{ϕ}}_{2} \oplus ϕ_{3}) \oplus ({\dot{ϕ}}_{2} \oplus ϕ_{3}) \oplus (ϕ_{2^{2}} \oplus {\dot{ϕ}}_{3^{2}})$

$≅ ϕ_{6} \oplus ϕ_{6} \oplus ϕ_{36}$

Therefore, trivially invariant factors of $ϕ_{6} \oplus ϕ_{6} \oplus ϕ_{36}$ are ${2, 6, 36}$ .

Solution of part (c)

Consider $ϕ_{10} \oplus ϕ_{20} \oplus ϕ_{30} \oplus ϕ_{40}$ .

The trivially factor of $ϕ_{10} \oplus ϕ_{20} \oplus ϕ_{30} \oplus ϕ_{40}$ is as follows:

$ϕ_{10} \oplus ϕ_{20} \oplus ϕ_{30} \oplus ϕ_{40} ≅ (ϕ_{2} \oplus ϕ_{5}) \oplus (ϕ_{2^{2}} \oplus ϕ_{5}) \oplus (ϕ_{2} \oplus ϕ_{3} \oplus ϕ_{5}) \oplus (α_{2^{3}} \oplus ϕ_{5})$

$≅ (ϕ_{2} \oplus ϕ_{5}) \oplus (ϕ_{2} \oplus ϕ_{5}) \oplus (ϕ_{2^{2}} \oplus ϕ_{5}) \oplus (ϕ_{2^{3}} \oplus ϕ_{3} \oplus ϕ_{5})$

$≅ ϕ_{10} \oplus ϕ_{10} \oplus ϕ_{20} \oplus ϕ_{120}$

Therefore, trivially invariant factors of role="math" localid="1657355395664" $ϕ_{10} \oplus ϕ_{10} \oplus ϕ_{20} \oplus ϕ_{120}$ are ${10, 10, 20, 120}$ .

Solution of part (d)

Consider $ϕ_{12} \oplus ϕ_{30} \oplus ϕ_{100} \oplus ϕ_{240}$ .

The trivially factor of $ϕ_{12} \oplus ϕ_{30} \oplus ϕ_{100} \oplus ϕ_{240}$ is as follows:

$ϕ_{12} \oplus ϕ_{30} \oplus ϕ_{100} \oplus ϕ_{240} ≅ (ϕ_{2}^{2} \oplus ϕ_{3}) \oplus (ϕ_{2} \oplus ϕ_{3} \oplus ϕ_{5}) \oplus ({\dot{ϕ}}_{2^{2}} \oplus ϕ_{5^{2}}) \oplus (ϕ_{2^{4}} \oplus ϕ_{3} \oplus ϕ_{5})$

$≅ ϕ_{2} \oplus (ϕ_{2} {}^{2}\oplus ϕ_{3} \oplus ϕ_{5}) \oplus (ϕ_{2^{2}} \oplus ϕ_{3} \oplus ϕ_{5}) \oplus (ϕ_{2^{2}} \oplus ϕ_{3} \oplus ϕ_{5^{2}})$

$≅ ϕ_{2} \oplus ϕ_{60} \oplus ϕ_{60} \oplus ϕ_{300}$

Therefore, trivially invariant factors $≅ ϕ_{2} \oplus ϕ_{60} \oplus ϕ_{60} \oplus ϕ_{300}$ of are ${2, 60, 60, 300}$ .

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

Short Answer

Step by step solution

Write the definition of invariant factors of G

Solution of part (a)

Solution of part (b)

Solution of part (c)

Solution of part (d)

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