Nilpotent, Abelian and Cyclic Numbers#

sympy.combinatorics.group_numbers.is_nilpotent_number(n)[source]#

Check whether \(n\) is a nilpotent number. A number \(n\) is said to be nilpotent if and only if every finite group of order \(n\) is nilpotent. For more information see [R43].

Examples

>>> from sympy.combinatorics.group_numbers import is_nilpotent_number
>>> from sympy import randprime
>>> is_nilpotent_number(21)
False
>>> is_nilpotent_number(randprime(1, 30)**12)
True

References

R43(1,2)

Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634.

sympy.combinatorics.group_numbers.is_abelian_number(n)[source]#

Check whether \(n\) is an abelian number. A number \(n\) is said to be abelian if and only if every finite group of order \(n\) is abelian. For more information see [R44].

Examples

>>> from sympy.combinatorics.group_numbers import is_abelian_number
>>> from sympy import randprime
>>> is_abelian_number(4)
True
>>> is_abelian_number(randprime(1, 2000)**2)
True
>>> is_abelian_number(60)
False

References

R44(1,2)

Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634.

sympy.combinatorics.group_numbers.is_cyclic_number(n)[source]#

Check whether \(n\) is a cyclic number. A number \(n\) is said to be cyclic if and only if every finite group of order \(n\) is cyclic. For more information see [R45].

Examples

>>> from sympy.combinatorics.group_numbers import is_cyclic_number
>>> from sympy import randprime
>>> is_cyclic_number(15)
True
>>> is_cyclic_number(randprime(1, 2000)**2)
False
>>> is_cyclic_number(4)
False

References

R45(1,2)

Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634.