List of all invariant factors (finite abelian groups)

asked 2024-07-07 22:34:25 +0100

po
45 ●5

updated 2024-07-07 22:35:18 +0100

I'm looking for a Sagemath builtin function giving the list of all the possible invariant factors of an abelian group with given finite order (say n)?

In Sagemath terminology, invariant factor is known as elementary divisor

For instance, if n = 48, it should return something like this:

[[2, 2, 2, 6], [2, 2, 12], [2, 24], [4, 12], [48]]

Or perhaps the list of all the possible elementary divisors in the proper sense.

This is not group theory but more or less combinatorics or counting.

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I could be wrong, but I don't think there is a built-in function. Using n=48 (for example) and n.factor()and the ideas at https://math.stackexchange.com/questi..., it shouldn't be hard to write something to do this.

John Palmieri ( 2024-07-08 00:27:37 +0100 )edit

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def elementary_divisors(n, m=1): if n == 1: yield [] return for d in n.divisors(): if d == 1 or d % m: continue for ll in elementary_divisors(n // d, d): yield [d] + ll

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Thanks for this code, it handles efficiently groups of order 2**50 (for instance), there are Partitions(50).cardinality() such groups.

po ( 2024-07-10 09:47:39 +0100 )edit

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List of all invariant factors (finite abelian groups)

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List of all invariant factors (finite abelian groups)