.. 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_02_solvable_groups.py"
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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_abstract_algebra_structure_plot_02_solvable_groups.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_abstract_algebra_structure_plot_02_solvable_groups.py:


Abel-Ruffini and solvable groups: why S_5 blocks the quintic
==================================================================

Computes the derived series of the symmetric groups S_2 to S_5. It
reaches the trivial group for n <= 4 but stalls at A_5 for n = 5, the
group-theoretic reason the general quintic has no formula in radicals.

.. GENERATED FROM PYTHON SOURCE LINES 11-13

.. code-block:: Python

    from mathematicskit.abstract_algebra import PermutationGroup, derived_series, is_solvable








.. GENERATED FROM PYTHON SOURCE LINES 14-16

Derived series of S_n
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python


    for n in range(2, 6):
        group = PermutationGroup(n)
        orders = [len(h) for h in derived_series(group)]
        print(f"S_{n}: derived series orders {orders} -> solvable: {is_solvable(group)}")





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

 .. code-block:: none

    S_2: derived series orders [2, 1] -> solvable: True
    S_3: derived series orders [6, 3, 1] -> solvable: True
    S_4: derived series orders [24, 12, 4, 1] -> solvable: True
    S_5: derived series orders [120, 60] -> solvable: False




.. GENERATED FROM PYTHON SOURCE LINES 23-29

The obstruction
-----------------------------------------------------

The series for S_5 stops at A_5 (order 60), which equals its own
commutator subgroup. By Galois's criterion, a degree-5 polynomial whose
Galois group is S_5 cannot be solved by radicals.


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

   **Total running time of the script:** (0 minutes 0.056 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_structure_plot_02_solvable_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_02_solvable_groups.ipynb <plot_02_solvable_groups.ipynb>`

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

      :download:`Download Python source code: plot_02_solvable_groups.py <plot_02_solvable_groups.py>`

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

      :download:`Download zipped: plot_02_solvable_groups.zip <plot_02_solvable_groups.zip>`


.. only:: html

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

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