.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/abstract_algebra/subgroups/plot_01_lagranges_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_subgroups_plot_01_lagranges_theorem.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_subgroups_plot_01_lagranges_theorem.py:


Lagrange's theorem: subgroup order divides group order
==============================================================

Enumerates every subgroup of Z_12 and confirms each subgroup's order
divides 12, and that its left cosets partition the whole group.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

.. code-block:: Python

    from mathematicskit.abstract_algebra import CyclicGroup, all_subgroups, cyclic_subgroup, left_cosets








.. GENERATED FROM PYTHON SOURCE LINES 13-15

Every subgroup of Z_12
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python


    g = CyclicGroup(12)
    for subgroup in sorted(all_subgroups(g), key=len):
        print(f"subgroup of order {len(subgroup)}: {sorted(subgroup)} (divides 12: {12 % len(subgroup) == 0})")





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

 .. code-block:: none

    subgroup of order 1: [0] (divides 12: True)
    subgroup of order 2: [0, 6] (divides 12: True)
    subgroup of order 3: [0, 4, 8] (divides 12: True)
    subgroup of order 4: [0, 3, 6, 9] (divides 12: True)
    subgroup of order 6: [0, 2, 4, 6, 8, 10] (divides 12: True)
    subgroup of order 12: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] (divides 12: True)




.. GENERATED FROM PYTHON SOURCE LINES 21-23

Cosets of the order-3 subgroup {0, 4, 8}
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-29

.. code-block:: Python


    h = cyclic_subgroup(g, 4)
    cosets = left_cosets(g, h)
    print(f"\nsubgroup H = {h}")
    print(f"cosets of H: {cosets}")
    print(f"index [G:H] = {len(cosets)} = |G|/|H| = {g.order}/{len(h)} = {g.order // len(h)}")




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

 .. code-block:: none


    subgroup H = [0, 4, 8]
    cosets of H: [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]]
    index [G:H] = 4 = |G|/|H| = 12/3 = 4





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_subgroups_plot_01_lagranges_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_lagranges_theorem.ipynb <plot_01_lagranges_theorem.ipynb>`

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

      :download:`Download Python source code: plot_01_lagranges_theorem.py <plot_01_lagranges_theorem.py>`

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

      :download:`Download zipped: plot_01_lagranges_theorem.zip <plot_01_lagranges_theorem.zip>`


.. only:: html

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

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