r"""Group homomorphisms: the homomorphism property, kernel, and image.
No numpy/scipy equivalent. See Dummit & Foote, *Abstract Algebra*, 3rd
ed., Sec. 1.6 and 3.3 (the isomorphism theorems), and E. Noether,
"Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und
Funktionenkörpern," Mathematische Annalen 96 (1927), 26-61.
"""
from __future__ import annotations
from typing import Callable
from mathematicskit.abstract_algebra.core.base import FiniteGroup, HomomorphismResult
__all__ = ["is_homomorphism", "homomorphism_kernel", "homomorphism_image", "analyze_homomorphism"]
[docs]
def is_homomorphism(source: FiniteGroup, target: FiniteGroup, phi: Callable) -> bool:
r"""Whether :math:`\varphi(ab) = \varphi(a)\varphi(b)` for every pair :math:`a, b \in G`.
Parameters
----------
source : FiniteGroup
target : FiniteGroup
phi : callable
Maps elements of `source` to elements of `target`.
Returns
-------
bool
Examples
--------
>>> from mathematicskit.abstract_algebra.systems.groups import CyclicGroup
>>> is_homomorphism(CyclicGroup(12), CyclicGroup(4), lambda a: a % 4)
True
>>> is_homomorphism(CyclicGroup(12), CyclicGroup(5), lambda a: a % 5)
False
"""
return all(phi(source.operate(a, b)) == target.operate(phi(a), phi(b)) for a in source.elements for b in source.elements)
[docs]
def homomorphism_kernel(source: FiniteGroup, target: FiniteGroup, phi: Callable) -> list:
r"""The kernel :math:`\ker\varphi = \{g \in G : \varphi(g) = e_H\}`, always a normal subgroup.
Parameters
----------
source : FiniteGroup
target : FiniteGroup
phi : callable
Returns
-------
list
Examples
--------
>>> from mathematicskit.abstract_algebra.systems.groups import CyclicGroup
>>> homomorphism_kernel(CyclicGroup(12), CyclicGroup(4), lambda a: a % 4)
[0, 4, 8]
"""
e = target.identity()
return [g for g in source.elements if phi(g) == e]
[docs]
def homomorphism_image(source: FiniteGroup, target: FiniteGroup, phi: Callable) -> list:
r"""The image :math:`\varphi(G) \le H`, in the target group's element order.
Parameters
----------
source : FiniteGroup
target : FiniteGroup
phi : callable
Returns
-------
list
Examples
--------
>>> from mathematicskit.abstract_algebra.systems.groups import CyclicGroup
>>> homomorphism_image(CyclicGroup(6), CyclicGroup(6), lambda a: (2 * a) % 6)
[0, 2, 4]
"""
values = {phi(g) for g in source.elements}
return [h for h in target.elements if h in values]
[docs]
def analyze_homomorphism(source: FiniteGroup, target: FiniteGroup, phi: Callable) -> HomomorphismResult:
r"""Kernel, image, and homomorphism check together.
The first isomorphism theorem, stated in its modern abstract form by
Emmy Noether in 1927, gives :math:`G/\ker\varphi \cong \varphi(G)`,
so :math:`|G| = |\ker\varphi| \cdot |\varphi(G)|`.
Parameters
----------
source : FiniteGroup
target : FiniteGroup
phi : callable
Returns
-------
HomomorphismResult
Examples
--------
>>> from mathematicskit.abstract_algebra.systems.groups import CyclicGroup
>>> result = analyze_homomorphism(CyclicGroup(12), CyclicGroup(4), lambda a: a % 4)
>>> len(result.kernel) * len(result.image) # = |Z_12|
12
"""
return HomomorphismResult(
kernel=homomorphism_kernel(source, target, phi),
image=homomorphism_image(source, target, phi),
is_homomorphism=is_homomorphism(source, target, phi),
)
