.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/abstract_algebra/structure/plot_01_cauchy_theorem.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_structure_plot_01_cauchy_theorem.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_structure_plot_01_cauchy_theorem.py:


Cauchy's theorem: elements of prime order
===============================================

For each prime p dividing a group's order, finds the elements of order
p, and checks McKay's refinement that their number is congruent to -1
modulo p.

.. GENERATED FROM PYTHON SOURCE LINES 11-15

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.abstract_algebra import DihedralGroup, PermutationGroup, QuaternionGroup, elements_of_order








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Elements of each prime order
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-28

.. code-block:: Python


    groups = {"S_4": PermutationGroup(4), "D_5": DihedralGroup(5), "Q_8": QuaternionGroup(), "S_5": PermutationGroup(5)}
    counts = {}
    for name, group in groups.items():
        for p in (2, 3, 5):
            if group.order % p == 0:
                count = len(elements_of_order(group, p))
                counts[(name, p)] = count
                print(f"{name} (order {group.order}): {count} elements of order {p}, count mod {p} = {count % p}")





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

 .. code-block:: none

    S_4 (order 24): 9 elements of order 2, count mod 2 = 1
    S_4 (order 24): 8 elements of order 3, count mod 3 = 2
    D_5 (order 10): 5 elements of order 2, count mod 2 = 1
    D_5 (order 10): 4 elements of order 5, count mod 5 = 4
    Q_8 (order 8): 1 elements of order 2, count mod 2 = 1
    S_5 (order 120): 25 elements of order 2, count mod 2 = 1
    S_5 (order 120): 20 elements of order 3, count mod 3 = 2
    S_5 (order 120): 24 elements of order 5, count mod 5 = 4




.. GENERATED FROM PYTHON SOURCE LINES 29-31

Every count is -1 mod p
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-39

.. code-block:: Python


    labels = [f"{name}, p={p}" for name, p in counts]
    fig, ax = plt.subplots(figsize=(8, 3.5))
    ax.bar(labels, list(counts.values()), color="tab:blue")
    ax.set_ylabel("elements of order p")
    ax.set_title("Cauchy's theorem: every prime divisor p of |G| has elements of order p")
    ax.tick_params(axis="x", rotation=30)
    fig.tight_layout()



.. image-sg:: /api/gallery/abstract_algebra/structure/images/sphx_glr_plot_01_cauchy_theorem_001.png
   :alt: Cauchy's theorem: every prime divisor p of |G| has elements of order p
   :srcset: /api/gallery/abstract_algebra/structure/images/sphx_glr_plot_01_cauchy_theorem_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.035 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_structure_plot_01_cauchy_theorem.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_cauchy_theorem.ipynb <plot_01_cauchy_theorem.ipynb>`

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

      :download:`Download Python source code: plot_01_cauchy_theorem.py <plot_01_cauchy_theorem.py>`

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

      :download:`Download zipped: plot_01_cauchy_theorem.zip <plot_01_cauchy_theorem.zip>`


.. only:: html

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

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