.. 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_04_quotient_groups.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_04_quotient_groups.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_structure_plot_04_quotient_groups.py:


Hölder's quotient groups: S_4 / V_4 is S_3
================================================

Finds the normal Klein four-group V_4 inside S_4, forms the quotient
S_4/V_4, and shows that it is a non-abelian group of order 6, like S_3.

.. GENERATED FROM PYTHON SOURCE LINES 10-13

.. code-block:: Python

    from mathematicskit.abstract_algebra import PermutationGroup, all_subgroups, is_normal_subgroup, quotient_group
    from mathematicskit.abstract_algebra.visualizers.plots import plot_cayley_table








.. GENERATED FROM PYTHON SOURCE LINES 14-16

Normal subgroups of S_4
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python


    s4 = PermutationGroup(4)
    normal = [h for h in all_subgroups(s4) if is_normal_subgroup(s4, h)]
    print(f"normal subgroup orders in S_4: {[len(h) for h in normal]}")





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

 .. code-block:: none

    normal subgroup orders in S_4: [1, 4, 12, 24]




.. GENERATED FROM PYTHON SOURCE LINES 22-24

The quotient by the Klein four-group
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-30

.. code-block:: Python


    v4 = next(h for h in normal if len(h) == 4)
    q = quotient_group(s4, v4)
    print(f"|S_4 / V_4| = {q.order}, abelian: {q.is_abelian()}")
    print(f"element orders in the quotient: {sorted(q.element_order(c) for c in q.elements)} (same as S_3)")
    plot_cayley_table(q)



.. image-sg:: /api/gallery/abstract_algebra/structure/images/sphx_glr_plot_04_quotient_groups_001.png
   :alt: Cayley table (order 6)
   :srcset: /api/gallery/abstract_algebra/structure/images/sphx_glr_plot_04_quotient_groups_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    |S_4 / V_4| = 6, abelian: False
    element orders in the quotient: [1, 2, 2, 2, 3, 3] (same as S_3)

    <Axes: title={'center': 'Cayley table (order 6)'}, xlabel='element index', ylabel='element index'>




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

   **Total running time of the script:** (0 minutes 0.128 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_structure_plot_04_quotient_groups.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_quotient_groups.ipynb <plot_04_quotient_groups.ipynb>`

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

      :download:`Download Python source code: plot_04_quotient_groups.py <plot_04_quotient_groups.py>`

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

      :download:`Download zipped: plot_04_quotient_groups.zip <plot_04_quotient_groups.zip>`


.. only:: html

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

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