.. _sphx_glr_api_gallery_abstract_algebra_homomorphisms:

Homomorphisms
-------------

Structure-preserving maps between groups, their kernels and images,
and the first isomorphism theorem.


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


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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Checks |G| = |\ker\varphi| \cdot |\operatorname{im}\varphi| for every homomorphism x \mapsto ax from Z_12 to itself, and shows that the sign map on S_4 has the alternating group A_4 as its kernel.">

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  .. image:: /api/gallery/abstract_algebra/homomorphisms/images/thumb/sphx_glr_plot_01_first_isomorphism_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/homomorphisms/plot_01_first_isomorphism_theorem`

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      <div class="sphx-glr-thumbnail-title">Noether's first isomorphism theorem</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/abstract_algebra/homomorphisms/plot_01_first_isomorphism_theorem