.. 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_05_composition_series.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_05_composition_series.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_structure_plot_05_composition_series.py:


Jordan-Hölder: composition series and their factors
=========================================================

Builds composition series for several groups and lists the orders of
their simple factors. Different groups of the same order can share the
same factors, and the Jordan-Hölder theorem says that every series of
one group has the same factors.

.. GENERATED FROM PYTHON SOURCE LINES 12-14

.. code-block:: Python

    from mathematicskit.abstract_algebra import CyclicGroup, DihedralGroup, PermutationGroup, composition_series








.. GENERATED FROM PYTHON SOURCE LINES 15-17

Composition factors
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. code-block:: Python


    groups = {"Z_12": CyclicGroup(12), "D_6": DihedralGroup(6), "S_4": PermutationGroup(4), "Z_24": CyclicGroup(24)}
    for name, group in groups.items():
        result = composition_series(group)
        orders = [len(h) for h in result.series]
        print(f"{name}: series orders {orders}, factor orders {result.factor_orders}")





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

 .. code-block:: none

    Z_12: series orders [12, 6, 3, 1], factor orders [2, 2, 3]
    D_6: series orders [12, 6, 3, 1], factor orders [2, 2, 3]
    S_4: series orders [24, 12, 4, 2, 1], factor orders [2, 3, 2, 2]
    Z_24: series orders [24, 12, 6, 3, 1], factor orders [2, 2, 2, 3]




.. GENERATED FROM PYTHON SOURCE LINES 25-31

Same factors, different groups
-----------------------------------------------------

Z_12 (abelian) and D_6 (non-abelian) are not isomorphic, yet both have
composition factors of orders 2, 2, 3. The Jordan-Hölder theorem makes
the factors an invariant of a group, but not a complete one.


.. 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_05_composition_series.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_05_composition_series.ipynb <plot_05_composition_series.ipynb>`

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

      :download:`Download Python source code: plot_05_composition_series.py <plot_05_composition_series.py>`

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

      :download:`Download zipped: plot_05_composition_series.zip <plot_05_composition_series.zip>`


.. only:: html

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

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