Problem 5.2.3 (Lists of elementary divisors)

Proof.

The answers are obtained with the method “ $\mathrm{abelian}\text{\_}\mathrm{groups}(n)$” given in the previous exercise (with the invariant factors in the second column).

(a)

$n=270=2\cdot 3^3\cdot 5$. $$\begin{array}{lllllll}\hfill & Z_2\times Z_{3^3}\times Z_5,& \hfill [2\cdot 3^3\cdot 5],& & \hfill & & \hfill \\ \hfill & Z_2\times Z_{3^2}\times Z_3\times Z_5,& \hfill [2\cdot 3^2\cdot 5,3],& & \hfill & & \hfill \\ \hfill & Z_2\times Z_3\times Z_3\times Z_3\times Z_5& \hfill [2\cdot 3\cdot 5,3,3].& & \hfill & & \hfill \end{array}$$

(b)

$n=9801=3^4\cdot 11^2$. $$\begin{array}{lllllll}\hfill & Z_{3^4}\times Z_{11^2}& \hfill [3^4\cdot 11^2]& & \hfill & & \hfill \\ \hfill & Z_{3^4}\times Z_{11}\times Z_{11}& \hfill [3^4\cdot 11,11]& & \hfill & & \hfill \\ \hfill & Z_{3^3}\times Z_3\times Z_{11^2}& \hfill [3^3\cdot 11^2,3]& & \hfill & & \hfill \\ \hfill & Z_{3^3}\times Z_3\times Z_{11}\times Z_{11}& \hfill [3^3\cdot 11,3\cdot 11]|& & \hfill & & \hfill \\ \hfill & Z_{3^2}\times Z_{3^2}\times Z_{11^2}& \hfill [3^2\cdot 11^2,3^2]& & \hfill & & \hfill \\ \hfill & Z_{3^2}\times Z_{3^2}\times Z_{11}\times Z_{11}& \hfill [3^2\cdot 11,3^2\cdot 11]& & \hfill & & \hfill \\ \hfill & Z_{3^2}\times Z_3\times Z_3\times Z_{11^2}& \hfill [3^2\cdot 11^2,3,3]& & \hfill & & \hfill \\ \hfill & Z_{3^2}\times Z_3\times Z_3\times Z_{11}\times Z_{11}& \hfill [3^2\cdot 11,3\cdot 11,3]& & \hfill & & \hfill \\ \hfill & Z_3\times Z_3\times Z_3\times Z_3\times Z_{11^2}& \hfill [3\cdot 11^2,3,3,3]& & \hfill & & \hfill \\ \hfill & Z_3\times Z_3\times Z_3\times Z_3\times Z_{11}\times Z_{11}& \hfill [3\cdot 11,3\cdot 11,3,3]& & \hfill & & \hfill \end{array}$$

(c)

$n=320=2^6\cdot 5$. $$\begin{array}{lllllll}\hfill & Z_{2^6}\times Z_5& \hfill [2^6\cdot 5]& & \hfill & & \hfill \\ \hfill & Z_{2^5}\times Z_2\times Z_5& \hfill [2^5\cdot 5,2]& & \hfill & & \hfill \\ \hfill & Z_{2^4}\times Z_2\times Z_2\times Z_5& \hfill [2^4\cdot 5,2^2]& & \hfill & & \hfill \\ \hfill & Z_{2^3}\times Z_{2^3}\times Z_5& \hfill [2^4\cdot 5,2,2]& & \hfill & & \hfill \\ \hfill & Z_{2^3}\times Z_{2^2}\times Z_2\times Z_5& \hfill [2^3\cdot 5,2^3]& & \hfill & & \hfill \\ \hfill & Z_2^3\times Z_2\times Z_2\times Z_2\times Z_5& \hfill [2^3\cdot 5,2,2,2]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_{2^2}\times Z_{2^2}\times Z_5& \hfill [2^2\cdot 5,2^2,2^2]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_{2^2}\times Z_2\times Z_2\times Z_5& \hfill [2^2\cdot 5,2^2,2,2]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_2\times Z_2\times Z_2\times Z_2\times Z_5& \hfill [2^2\cdot 5,2,2,2,2]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_2\times Z_2\times Z_2\times Z_2\times Z_5& \hfill [2\cdot 5,2,2,2,2,2]& & \hfill & & \hfill \end{array}$$

(d)

$n=105=3\cdot 5\cdot 7$.

$$\begin{array}{lllllll}\hfill Z_3\times Z_5\times Z_7& & \hfill [3\cdot 5\cdot 7]& & \hfill & & \hfill \end{array}$$

(e)

$n=44100=2^2\cdot 3^2\cdot 5^2\cdot 7^2$ $$\begin{array}{lllllll}\hfill & Z_{2^2}\times Z_{3^2}\times Z_{5^2}\times Z_{7^2}& \hfill [2^2\cdot 3^2\cdot 5^2\cdot 7^2]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_{3^2}\times Z_{5^2}\times Z_7\times Z_7& \hfill [2^2\cdot 3^2\cdot 5^2\cdot 7,7]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_{3^2}\times Z_5\times Z_5\times Z_{7^2}& \hfill [2^2\cdot 3^2\cdot 5\cdot 7^2,5]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_{3^2}\times Z_5\times Z_5\times Z_7\times Z_7& \hfill [2^2\cdot 3^2\cdot 5\cdot 7,5\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_3\times Z_3\times Z_{5^2}\times Z_7^2& \hfill [2^2\cdot 3\cdot 5^2\cdot 7^2,3]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_3\times Z_3\times Z_{5^2}\times Z_7\times Z_7& \hfill [2^2\cdot 3\cdot 5^2\cdot 7,3\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_3\times Z_3\times Z_5\times Z_5\times Z_{7^2}& \hfill [2^2\cdot 3\cdot 5\cdot 7^2,3\cdot 5]& & \hfill & & \hfill \\ \hfill & Z_{2^2}\times Z_3\times Z_3\times Z_5\times Z_5\times Z_7\times Z_7& \hfill [2^2\cdot 3\cdot 5\cdot 7,3\cdot 5\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_{3^2}\times Z_{5^2}\times Z_{7^2}& \hfill [2\cdot 3^2\cdot 5^2\cdot 7^2,2]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_{3^2}\times Z_{5^2}\times Z_7\times Z_7& \hfill [2\cdot 3^2\cdot 5^2\cdot 7,2\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_{3^2}\times Z_5\times Z_5\times Z_{7^2}& \hfill [2\cdot 3^2\cdot 5\cdot 7^2,2\cdot 5]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_{3^2}\times Z_5\times Z_5\times Z_7\times Z_7& \hfill [2\cdot 3^2\cdot 5\cdot 7,2\cdot 5\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_3\times Z_3\times Z_{5^2}\times Z_{7^2}& \hfill [2\cdot 3\cdot 5^2\cdot 7^2,2\cdot 3]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_3\times Z_3\times Z_{5^2}\times Z_7\times Z_7& \hfill [2\cdot 3\cdot 5^2\cdot 7,2\cdot 3\cdot 7]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_3\times Z_3\times Z_5\times Z_5\times Z_{7^2}& \hfill [2\cdot 3\cdot 5\cdot 7^2,2\cdot 3\cdot 5]& & \hfill & & \hfill \\ \hfill & Z_2\times Z_2\times Z_3\times Z_3\times Z_5\times Z_5\times Z_7\times Z_7& \hfill [2\cdot 3\cdot 5\cdot 7,2\cdot 3\cdot 5\cdot 7]& & \hfill & & \hfill \end{array}$$