Number of groups¶
- sympy.combinatorics.group_numbers.is_nilpotent_number(n) bool [source]¶
Check whether
is a nilpotent number. A number
is said to be nilpotent if and only if every finite group of order
is nilpotent. For more information see [R48].
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
- sympy.combinatorics.group_numbers.is_abelian_number(n) bool [source]¶
Check whether
is an abelian number. A number
is said to be abelian if and only if every finite group of order
is abelian. For more information see [R50].
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
- sympy.combinatorics.group_numbers.is_cyclic_number(n) bool [source]¶
Check whether
is a cyclic number. A number
is said to be cyclic if and only if every finite group of order
is cyclic. For more information see [R52].
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
- sympy.combinatorics.group_numbers.groups_count(n)[source]¶
Number of groups of order
. In [R54],
gnu(n)
is given, so we follow this notation here as well.- Parameters:
n : Integer
n
is a positive integer- Returns:
int :
gnu(n)
- Raises:
ValueError
Number of groups of order
n
is unknown or not implemented. For example, gnu() is not yet known. On the other hand, gnu(12) is known to be 5, but this has not yet been implemented in this function.
Examples
>>> from sympy.combinatorics.group_numbers import groups_count >>> groups_count(3) # There is only one cyclic group of order 3 1 >>> # There are two groups of order 10: the cyclic group and the dihedral group >>> groups_count(10) 2
See also
is_cyclic_number
is cyclic iff gnu(n) = 1
References
[R54] (1,2)John H. Conway, Heiko Dietrich and E.A. O’Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731