.. 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_03_sylow_subgroups.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_03_sylow_subgroups.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_structure_plot_03_sylow_subgroups.py:


Sylow's theorems in S_4
=============================

Finds the Sylow 2- and 3-subgroups of the symmetric group S_4 (order
24 = 2^3 * 3) and checks the counting constraints n_p = 1 (mod p) and
:math:`n_p \mid |G|/p^k`.

.. GENERATED FROM PYTHON SOURCE LINES 11-13

.. code-block:: Python

    from mathematicskit.abstract_algebra import PermutationGroup, sylow_subgroups








.. GENERATED FROM PYTHON SOURCE LINES 14-16

Sylow subgroups of S_4
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python


    s4 = PermutationGroup(4)
    for p in (2, 3):
        result = sylow_subgroups(s4, p)
        m = s4.order // result.sylow_order
        print(f"p = {p}: Sylow order {result.sylow_order}, n_{p} = {result.count}")
        print(f"  n_p mod p = {result.count % p} (should be 1), n_p divides {m}: {m % result.count == 0}")
        for h in result.subgroups:
            print(f"    {h}")




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

 .. code-block:: none

    p = 2: Sylow order 8, n_2 = 3
      n_p mod p = 1 (should be 1), n_p divides 3: True
        [(0, 1, 2, 3), (0, 1, 3, 2), (1, 0, 2, 3), (1, 0, 3, 2), (2, 3, 0, 1), (2, 3, 1, 0), (3, 2, 0, 1), (3, 2, 1, 0)]
        [(0, 1, 2, 3), (0, 3, 2, 1), (1, 0, 3, 2), (1, 2, 3, 0), (2, 1, 0, 3), (2, 3, 0, 1), (3, 0, 1, 2), (3, 2, 1, 0)]
        [(0, 1, 2, 3), (0, 2, 1, 3), (1, 0, 3, 2), (1, 3, 0, 2), (2, 0, 3, 1), (2, 3, 0, 1), (3, 1, 2, 0), (3, 2, 1, 0)]
    p = 3: Sylow order 3, n_3 = 4
      n_p mod p = 1 (should be 1), n_p divides 8: True
        [(0, 1, 2, 3), (0, 2, 3, 1), (0, 3, 1, 2)]
        [(0, 1, 2, 3), (2, 1, 3, 0), (3, 1, 0, 2)]
        [(0, 1, 2, 3), (1, 2, 0, 3), (2, 0, 1, 3)]
        [(0, 1, 2, 3), (1, 3, 2, 0), (3, 0, 2, 1)]





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

   **Total running time of the script:** (0 minutes 0.208 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_structure_plot_03_sylow_subgroups.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_sylow_subgroups.ipynb <plot_03_sylow_subgroups.ipynb>`

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

      :download:`Download Python source code: plot_03_sylow_subgroups.py <plot_03_sylow_subgroups.py>`

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

      :download:`Download zipped: plot_03_sylow_subgroups.zip <plot_03_sylow_subgroups.zip>`


.. only:: html

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

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