Nilpotent, abelian and cyclic numbers

[giorgosgiapis]

Add methods to check whether a number is nilpotent, abelian, or cyclic. A number n is called nilpotent (abelian/cyclic) if every group of order n is nilpotent (abelian/cyclic). More details about these numbers can be found here: http://www2.math.ou.edu/~shankar/papers/nil2.pdf.

Component: group theory

Keywords: nilpotent, abelian, cyclic

Author: Georgios Giapitzakis Tzintanos

Branch/Commit: u/gh-giorgosgiapis/nilpotent_numbers @ d0e5576

Issue created by migration from https://trac.sagemath.org/ticket/32464

[giorgosgiapis]

Changed keywords from none to nilpotent, abelian, cyclic

[giorgosgiapis]

Description changed:

--- 
+++ 
@@ -1 +1 @@
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+Add methods to check whether a number is nilpotent, abelian, or cyclic. A number `n` is called nilpotent (abelian/cyclic) if every group of order `n` is nilpotent (abelian/cyclic). More details about these numbers can be found here: [http://www2.math.ou.edu/~shankar/papers/nil2.pdf](http://www2.math.ou.edu/~shankar/papers/nil2.pdf).

[giorgosgiapis]

Branch: u/gh-giorgosgiapis/nilpotent_numbers

[giorgosgiapis]

New commits:

d0e5576

Added checks for nilpotent, abelian and cyclic numbers

[giorgosgiapis]

Commit: d0e5576

[giorgosgiapis]

Author: Georgios Giapitzakis Tzintanos

[mkoeppe]

comment:5

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.

[DaveWitteMorris]

comment:7

I like the idea, but I think it would be much more useful to have a single function that combines all three of these (and does more). Perhaps the function should be:

exists_group_of_order(
    n, 
    *,  # this means that the remaining arguments are keyword-only
    cyclic=None, 
    abelian=None, 
    nilpotent=None,
    solvable=None,
    simple=None
    )

For example:

• exists_group_of_order(n) would be True if n is any integer that is at least 1, because there does exist a group of order n.
• exists_group_of_order(n, abelian=False) would be True if and only if n is not an abelian number.
• exists_group_of_order(n, abelian=False, solvable=True) would be True if and only if there is a nonabelian solvable group of order n.
• exists_group_of_order(n, nilpotent=False, solvable=True) would be False for all values of n, because all solvable groups are nilpotent.
• exists_group_of_order(60, cyclic=False, simple=True) would be True because there is a noncyclic simple group of order 60.

Or perhaps this should be a method of the Integer class that is named is_order_of_group. (Other keywords could be added, such as perfect, but I think the five that I already named are the most important.)

[DaveWitteMorris]

comment:8

FYI: Methods for nilpotent, abelian and cyclic numbers were added to sympy in version 1.11 https://github.com/sympy/sympy/pull/23329.