.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/abstract_algebra/groups/plot_04_cayley_tables.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_abstract_algebra_groups_plot_04_cayley_tables.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_abstract_algebra_groups_plot_04_cayley_tables.py:


Cayley's abstract group and its multiplication table
==========================================================

Cayley's 1854 definition treats a group purely as symbols with a
binary operation. Here the group of his title, generated by
:math:`\theta` with :math:`\theta^n = 1`, is written as an abstract
:class:`~mathematicskit.abstract_algebra.core.base.FiniteGroup` whose
elements are just the strings ``"1", "θ", "θ²", ...``. Its Cayley table
is identical to that of :math:`\mathbb{Z}_6`, while the non-abelian
:math:`S_3` of the same order gives a visibly different, non-symmetric
table.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.abstract_algebra import CyclicGroup, FiniteGroup, PermutationGroup








.. GENERATED FROM PYTHON SOURCE LINES 22-24

A group defined only by the symbolic equation theta^n = 1
-----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-56

.. code-block:: Python



    class ThetaGroup(FiniteGroup):
        """Cayley's group 1, θ, θ², ..., θ^(n-1) with θ^n = 1, as bare symbols."""

        def __init__(self, n):
            self.n = n
            self._symbols = ["1", "θ"] + [f"θ^{k}" for k in range(2, n)]

        @property
        def elements(self):
            return self._symbols

        def identity(self):
            return "1"

        def operate(self, a, b):
            return self._symbols[(self._symbols.index(a) + self._symbols.index(b)) % self.n]

        def inverse(self, a):
            return self._symbols[-self._symbols.index(a) % self.n]


    theta6 = ThetaGroup(6)
    z6 = CyclicGroup(6)
    s3 = PermutationGroup(3)
    print(f"theta-group order {theta6.order}, abelian: {theta6.is_abelian()}")
    print(f"theta^6 = 1: {theta6.element_order('θ') == 6}")
    print(f"same Cayley table as Z_6: {np.array_equal(theta6.cayley_table(), z6.cayley_table())}")
    print(f"S_3 table symmetric: {np.array_equal(s3.cayley_table(), s3.cayley_table().T)}")






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    theta-group order 6, abelian: True
    theta^6 = 1: True
    same Cayley table as Z_6: True
    S_3 table symmetric: False




.. GENERATED FROM PYTHON SOURCE LINES 57-59

Cayley tables, labelled with the element names
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 59-80

.. code-block:: Python



    def labelled_table(ax, group, labels, title):
        table = group.cayley_table()
        ax.imshow(table, cmap="tab10", vmin=0, vmax=9)
        for i in range(group.order):
            for j in range(group.order):
                ax.text(j, i, labels[table[i, j]], ha="center", va="center", fontsize=8)
        ax.set_xticks(range(group.order), labels, fontsize=8, rotation=45)
        ax.set_yticks(range(group.order), labels, fontsize=8)
        ax.xaxis.tick_top()
        ax.set_title(title, pad=28)


    fig, axes = plt.subplots(1, 3, figsize=(14, 5))
    theta_labels = ["1", r"$\theta$"] + [rf"$\theta^{k}$" for k in range(2, 6)]
    labelled_table(axes[0], theta6, theta_labels, r"Cayley's $\theta^6 = 1$")
    labelled_table(axes[1], z6, [str(k) for k in z6.elements], r"$\mathbb{Z}_6$ (same pattern)")
    s3_labels = ["".join(map(str, p)) for p in s3.elements]
    labelled_table(axes[2], s3, s3_labels, r"$S_3$ (not symmetric: non-abelian)")
    fig.tight_layout()



.. image-sg:: /api/gallery/abstract_algebra/groups/images/sphx_glr_plot_04_cayley_tables_001.png
   :alt: Cayley's $\theta^6 = 1$, $\mathbb{Z}_6$ (same pattern), $S_3$ (not symmetric: non-abelian)
   :srcset: /api/gallery/abstract_algebra/groups/images/sphx_glr_plot_04_cayley_tables_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.107 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_groups_plot_04_cayley_tables.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_04_cayley_tables.ipynb <plot_04_cayley_tables.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_04_cayley_tables.py <plot_04_cayley_tables.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_cayley_tables.zip <plot_04_cayley_tables.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_