Problem 5.2.2 (List of invariant factors)

Proof. Answers:

(a)

$n=270=2\cdot 3^3\cdot 5$.

Invariant factors:

$$\begin{array}{llll}\hfill & [2\cdot 3^3\cdot 5]& \hfill & \\ \hfill & [2\cdot 3^2\cdot 5,3]& \hfill & \\ \hfill & [2\cdot 3\cdot 5,3,3]& \hfill & \end{array}$$

(b)

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

(c)

$n=320=2^6\cdot 5$.

$$\begin{array}{llll}\hfill & [2^6\cdot 5]& \hfill & \\ \hfill & [2^5\cdot 5,2]& \hfill & \\ \hfill & [2^4\cdot 5,2^2]& \hfill & \\ \hfill & [2^4\cdot 5,2,2]& \hfill & \\ \hfill & [2^3\cdot 5,2^3]& \hfill & \\ \hfill & [2^3\cdot 5,2^2,2]& \hfill & \\ \hfill & [2^3\cdot 5,2,2,2]& \hfill & \\ \hfill & [2^2\cdot 5,2^2,2^2]& \hfill & \\ \hfill & [2^2\cdot 5,2^2,2,2]& \hfill & \\ \hfill & [2^2\cdot 5,2,2,2,2]& \hfill & \\ \hfill & [2\cdot 5,2,2,2,2,2]& \hfill & \end{array}$$

(d)

$n=105=3\cdot 5\cdot 7$. $$\begin{array}{llll}\hfill & [3\cdot 5\cdot 7]& \hfill & \end{array}$$

(e)

$n=44100=2^2\cdot 3^2\cdot 5^2\cdot 7^2$.

$$\begin{array}{llll}\hfill & [2^2\cdot 3^2\cdot 5^2\cdot 7^2]& \hfill & \\ \hfill & [2^2\cdot 3^2\cdot 5^2\cdot 7,7]& \hfill & \\ \hfill & [2^2\cdot 3^2\cdot 5\cdot 7^2,5]& \hfill & \\ \hfill & [2^2\cdot 3^2\cdot 5\cdot 7,5\cdot 7]& \hfill & \\ \hfill & [2^2\cdot 3\cdot 5^2\cdot 7^2,3]& \hfill & \\ \hfill & [2^2\cdot 3\cdot 5^2\cdot 7,3\cdot 7]& \hfill & \\ \hfill & [2^2\cdot 3\cdot 5\cdot 7^2,3\cdot 5]& \hfill & \\ \hfill & [2^2\cdot 3\cdot 5\cdot 7,3\cdot 5\cdot 7]& \hfill & \\ \hfill & [2\cdot 3^2\cdot 5^2\cdot 7^2,2]& \hfill & \\ \hfill & [2\cdot 3^2\cdot 5^2\cdot 7,2\cdot 7]& \hfill & \\ \hfill & [2\cdot 3^2\cdot 5\cdot 7^2,2\cdot 5]& \hfill & \\ \hfill & [2\cdot 3^2\cdot 5\cdot 7,2\cdot 5\cdot 7]& \hfill & \\ \hfill & [2\cdot 3\cdot 5^2\cdot 7^2,2\cdot 3]& \hfill & \\ \hfill & [2\cdot 3\cdot 5^2\cdot 7,2\cdot 3\cdot 7]& \hfill & \\ \hfill & [2\cdot 3\cdot 5\cdot 7^2,2\cdot 3\cdot 5]& \hfill & \\ \hfill & [2\cdot 3\cdot 5\cdot 7,2\cdot 3\cdot 5\cdot 7]& \hfill & \end{array}$$

I obtain these results with the following homemade code in Sagemath:

sage: def abelian_groups(n):
....:     l = []
....:     for p, a in n.factor():
....:         l1 = [[p, list(u)] for u in Partitions(a).list()]
....:         if l == []:
....:             l2 = []
....:             for y in l1:
....:                 l2.append([y])
....:         else:
....:             l2 = []
....:             for x in l:
....:                 for y in l1:
....:                     u = copy(x)
....:                     u.append(y)
....:                     l2.append(u)
....:         l = l2
....:     return l
....:
sage: def invariant_factors(group):
....:     maxi = 0
....:     for it in group:
....:         if len(it[1])>maxi:
....:             maxi = len(it[1])
....:     gr = []
....:     for p,it in group:
....:         expo = copy(it)
....:         for i in range(maxi - len(it)):
....:             expo.append(0)
....:         gr.append([p, expo])
....:     l = []
....:     for i in range(maxi):
....:         inv_fact = 1
....:         for p, it in gr:
....:             inv_fact *= p^it[i]
....:         l.append(inv_fact.factor())
....:     return l
....:
sage: n = 270
sage: for gr in abelian_groups(n):
....:     print(invariant_factors(gr))
....:
                                                                  

                                                                  
[2 * 3^3 * 5]
[2 * 3^2 * 5, 3]
[2 * 3 * 5, 3, 3]
sage:
sage: n = 9801
sage: for gr in abelian_groups(n):
....:     print(invariant_factors(gr))
....:
[3^4 * 11^2]
[3^4 * 11, 11]
[3^3 * 11^2, 3]
[3^3 * 11, 3 * 11]
[3^2 * 11^2, 3^2]
[3^2 * 11, 3^2 * 11]
[3^2 * 11^2, 3, 3]
[3^2 * 11, 3 * 11, 3]
[3 * 11^2, 3, 3, 3]
[3 * 11, 3 * 11, 3, 3]
sage:
sage: n = 320
sage: for gr in abelian_groups(n):
....:     print(invariant_factors(gr))
....:
[2^6 * 5]
[2^5 * 5, 2]
[2^4 * 5, 2^2]
[2^4 * 5, 2, 2]
[2^3 * 5, 2^3]
[2^3 * 5, 2^2, 2]
[2^3 * 5, 2, 2, 2]
[2^2 * 5, 2^2, 2^2]
[2^2 * 5, 2^2, 2, 2]
[2^2 * 5, 2, 2, 2, 2]
[2 * 5, 2, 2, 2, 2, 2]
sage:
sage: n = 105
sage: for gr in abelian_groups(n):
....:     print(invariant_factors(gr))
....:
[3 * 5 * 7]
sage:
sage: n = 44100
sage: for gr in abelian_groups(n):
....:     print(invariant_factors(gr))
....:
[2^2 * 3^2 * 5^2 * 7^2]
                                                                  

                                                                  
[2^2 * 3^2 * 5^2 * 7, 7]
[2^2 * 3^2 * 5 * 7^2, 5]
[2^2 * 3^2 * 5 * 7, 5 * 7]
[2^2 * 3 * 5^2 * 7^2, 3]
[2^2 * 3 * 5^2 * 7, 3 * 7]
[2^2 * 3 * 5 * 7^2, 3 * 5]
[2^2 * 3 * 5 * 7, 3 * 5 * 7]
[2 * 3^2 * 5^2 * 7^2, 2]
[2 * 3^2 * 5^2 * 7, 2 * 7]
[2 * 3^2 * 5 * 7^2, 2 * 5]
[2 * 3^2 * 5 * 7, 2 * 5 * 7]
[2 * 3 * 5^2 * 7^2, 2 * 3]
[2 * 3 * 5^2 * 7, 2 * 3 * 7]
[2 * 3 * 5 * 7^2, 2 * 3 * 5]
[2 * 3 * 5 * 7, 2 * 3 * 5 * 7]

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