.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/abstract_algebra/groups/plot_03_quaternions.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_groups_plot_03_quaternions.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_groups_plot_03_quaternions.py:


Hamilton's quaternions: the group Q_8
===========================================

Verifies Hamilton's relations i^2 = j^2 = k^2 = ijk = -1 in the
quaternion group, shows that it is non-abelian, and prints its
multiplication table.

.. GENERATED FROM PYTHON SOURCE LINES 11-14

.. code-block:: Python

    from mathematicskit.abstract_algebra import QuaternionGroup, all_subgroups, is_normal_subgroup
    from mathematicskit.abstract_algebra.visualizers.plots import plot_cayley_table








.. GENERATED FROM PYTHON SOURCE LINES 15-17

Hamilton's relations
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. code-block:: Python


    q = QuaternionGroup()
    for unit in ("i", "j", "k"):
        print(f"{unit}^2 = {q.operate(unit, unit)}")
    print(f"ijk = {q.operate(q.operate('i', 'j'), 'k')}")
    print(f"ij = {q.operate('i', 'j')}, but ji = {q.operate('j', 'i')}")





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

 .. code-block:: none

    i^2 = -1
    j^2 = -1
    k^2 = -1
    ijk = -1
    ij = k, but ji = -k




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Multiplication table
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-32

.. code-block:: Python


    print("\n      " + "".join(f"{b:>4}" for b in q.elements))
    for a in q.elements:
        print(f"{a:>4}  " + "".join(f"{q.operate(a, b):>4}" for b in q.elements))





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

 .. code-block:: none


             1  -1   i  -i   j  -j   k  -k
       1     1  -1   i  -i   j  -j   k  -k
      -1    -1   1  -i   i  -j   j  -k   k
       i     i  -i  -1   1   k  -k  -j   j
      -i    -i   i   1  -1  -k   k   j  -j
       j     j  -j  -k   k  -1   1   i  -i
      -j    -j   j   k  -k   1  -1  -i   i
       k     k  -k   j  -j  -i   i  -1   1
      -k    -k   k  -j   j   i  -i   1  -1




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Non-abelian, yet every subgroup is normal
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-39

.. code-block:: Python


    subgroups = all_subgroups(q)
    print(f"\n{len(subgroups)} subgroups, all normal: {all(is_normal_subgroup(q, h) for h in subgroups)}")
    plot_cayley_table(q)



.. image-sg:: /api/gallery/abstract_algebra/groups/images/sphx_glr_plot_03_quaternions_001.png
   :alt: Cayley table (order 8)
   :srcset: /api/gallery/abstract_algebra/groups/images/sphx_glr_plot_03_quaternions_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    6 subgroups, all normal: True

    <Axes: title={'center': 'Cayley table (order 8)'}, xlabel='element index', ylabel='element index'>




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

   **Total running time of the script:** (0 minutes 0.023 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_groups_plot_03_quaternions.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_quaternions.ipynb <plot_03_quaternions.ipynb>`

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

      :download:`Download Python source code: plot_03_quaternions.py <plot_03_quaternions.py>`

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

      :download:`Download zipped: plot_03_quaternions.zip <plot_03_quaternions.zip>`


.. only:: html

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

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