.. _sphx_glr_api_gallery_abstract_algebra_structure:

Group structure
---------------

Element orders, normal subgroups and quotients, solvability, Sylow
subgroups, and composition series.


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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="For each prime p dividing a group&#x27;s order, finds the elements of order p, and checks McKay&#x27;s refinement that their number is congruent to -1 modulo p.">

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  .. image:: /api/gallery/abstract_algebra/structure/images/thumb/sphx_glr_plot_01_cauchy_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/structure/plot_01_cauchy_theorem`

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      <div class="sphx-glr-thumbnail-title">Cauchy's theorem: elements of prime order</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Computes the derived series of the symmetric groups S_2 to S_5. It reaches the trivial group for n &lt;= 4 but stalls at A_5 for n = 5, the group-theoretic reason the general quintic has no formula in radicals.">

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  .. image:: /api/gallery/abstract_algebra/structure/images/thumb/sphx_glr_plot_02_solvable_groups_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/structure/plot_02_solvable_groups`

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      <div class="sphx-glr-thumbnail-title">Abel-Ruffini and solvable groups: why S_5 blocks the quintic</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Finds the Sylow 2- and 3-subgroups of the symmetric group S_4 (order 24 = 2^3 * 3) and checks the counting constraints n_p = 1 (mod p) and n_p \mid |G|/p^k.">

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  .. image:: /api/gallery/abstract_algebra/structure/images/thumb/sphx_glr_plot_03_sylow_subgroups_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/structure/plot_03_sylow_subgroups`

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      <div class="sphx-glr-thumbnail-title">Sylow's theorems in S_4</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Finds the normal Klein four-group V_4 inside S_4, forms the quotient S_4/V_4, and shows that it is a non-abelian group of order 6, like S_3.">

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  .. image:: /api/gallery/abstract_algebra/structure/images/thumb/sphx_glr_plot_04_quotient_groups_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/structure/plot_04_quotient_groups`

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      <div class="sphx-glr-thumbnail-title">Hölder's quotient groups: S_4 / V_4 is S_3</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="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.">

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  .. image:: /api/gallery/abstract_algebra/structure/images/thumb/sphx_glr_plot_05_composition_series_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/structure/plot_05_composition_series`

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      <div class="sphx-glr-thumbnail-title">Jordan-Hölder: composition series and their factors</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/abstract_algebra/structure/plot_01_cauchy_theorem
   /api/gallery/abstract_algebra/structure/plot_02_solvable_groups
   /api/gallery/abstract_algebra/structure/plot_03_sylow_subgroups
   /api/gallery/abstract_algebra/structure/plot_04_quotient_groups
   /api/gallery/abstract_algebra/structure/plot_05_composition_series