.. _sphx_glr_api_gallery_abstract_algebra_groups:

Groups
--------

Permutation groups of polynomial roots, dihedral and quaternion groups,
Cayley tables, and group-property checks.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Galois turned the question &quot;is this equation solvable?&quot; into a question about a group of permutations of its roots. The roots of x^3 - 2 are \sqrt[3]{2}, \sqrt[3]{2}\,\omega and \sqrt[3]{2}\,\omega^2 (with \omega = e^{2\pi i/3}). Complex conjugation swaps the two non-real roots, multiplication by \omega cycles all three, and closing these two permutations under composition gives Galois&#x27;s group -- the full symmetric group S_3, which is non-abelian.">

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  .. image:: /api/gallery/abstract_algebra/groups/images/thumb/sphx_glr_plot_01_galois_permutation_groups_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/groups/plot_01_galois_permutation_groups`

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      <div class="sphx-glr-thumbnail-title">Galois's permutation groups: S_3 permuting the roots of x^3 - 2</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Builds the dihedral group D_4 of the eight symmetries of a square, checks its defining relations, and draws its Cayley table.">

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  .. image:: /api/gallery/abstract_algebra/groups/images/thumb/sphx_glr_plot_02_dihedral_symmetries_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/groups/plot_02_dihedral_symmetries`

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      <div class="sphx-glr-thumbnail-title">Klein's Erlangen program: the symmetry group of a square</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Verifies Hamilton&#x27;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.">

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  .. image:: /api/gallery/abstract_algebra/groups/images/thumb/sphx_glr_plot_03_quaternions_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/groups/plot_03_quaternions`

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      <div class="sphx-glr-thumbnail-title">Hamilton's quaternions: the group Q_8</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Cayley&#x27;s 1854 definition treats a group purely as symbols with a binary operation. Here the group of his title, generated by \theta with \theta^n = 1, is written as an abstract FiniteGroup whose elements are just the strings &quot;1&quot;, &quot;θ&quot;, &quot;θ²&quot;, .... Its Cayley table is identical to that of \mathbb{Z}_6, while the non-abelian S_3 of the same order gives a visibly different, non-symmetric table.">

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  .. image:: /api/gallery/abstract_algebra/groups/images/thumb/sphx_glr_plot_04_cayley_tables_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/groups/plot_04_cayley_tables`

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      <div class="sphx-glr-thumbnail-title">Cayley's abstract group and its multiplication table</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/abstract_algebra/groups/plot_01_galois_permutation_groups
   /api/gallery/abstract_algebra/groups/plot_02_dihedral_symmetries
   /api/gallery/abstract_algebra/groups/plot_03_quaternions
   /api/gallery/abstract_algebra/groups/plot_04_cayley_tables