:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.abstract_algebra``:
cyclic and permutation groups, subgroups and cosets, finite fields, and
polynomial ring arithmetic.

Each script in this gallery is self-contained and can be run directly with
``python examples/abstract_algebra/<section>/<script>.py``.

Sections
--------

- **groups** -- cyclic and permutation groups, Cayley tables, and
  group-property checks.
- **subgroups** -- subgroup and coset enumeration, and Lagrange's
  theorem.
- **finite_fields** -- ``GF(p)`` and ``GF(p^n)`` arithmetic.
- **polynomial_ring** -- polynomial addition, multiplication, division,
  and gcd.


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Group actions
-------------

Orbits of a group acting on a finite set, and Burnside's orbit-counting
lemma.


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    <div class="sphx-glr-thumbcontainer" tooltip="Counts two-colored necklaces (up to rotation) and bracelets (up to rotation and reflection) by averaging fixed points, and checks each count against direct orbit enumeration.">

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  .. image:: /api/gallery/abstract_algebra/actions/images/thumb/sphx_glr_plot_01_burnside_necklaces_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/actions/plot_01_burnside_necklaces`

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      <div class="sphx-glr-thumbnail-title">Burnside's lemma: counting necklaces</div>
    </div>


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Error-correcting codes
----------------------

Reed-Solomon codes: polynomial evaluation over a finite field, and
recovery from erasures.


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    <div class="sphx-glr-thumbcontainer" tooltip="Encodes a short message as the values of a polynomial over GF(929), erases symbols at random, and recovers the message from any k surviving values.">

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  .. image:: /api/gallery/abstract_algebra/codes/images/thumb/sphx_glr_plot_01_reed_solomon_erasures_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/codes/plot_01_reed_solomon_erasures`

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      <div class="sphx-glr-thumbnail-title">Reed-Solomon codes: recovering from erasures</div>
    </div>


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Finite fields
---------------

``GF(p)`` and ``GF(p^n)`` arithmetic.


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    <div class="sphx-glr-thumbcontainer" tooltip="Finds a degree-3 irreducible polynomial over GF(2), builds the field GF(2^3) = GF(8) from it, and verifies every nonzero element has a multiplicative inverse.">

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  .. image:: /api/gallery/abstract_algebra/finite_fields/images/thumb/sphx_glr_plot_01_gf8_multiplication_table_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/finite_fields/plot_01_gf8_multiplication_table`

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      <div class="sphx-glr-thumbnail-title">GF(8): finding an irreducible polynomial and building the field</div>
    </div>


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Groups
--------

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


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

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


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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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Polynomial ring arithmetic
-----------------------------

Polynomial addition, multiplication, division, and gcd.


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    <div class="sphx-glr-thumbcontainer" tooltip="Divides two polynomials with remainder, and finds their greatest common divisor via the polynomial Euclidean algorithm -- exactly the integer Euclidean algorithm with polynomial division in place of integer division.">

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  .. image:: /api/gallery/abstract_algebra/polynomial_ring/images/thumb/sphx_glr_plot_01_division_and_gcd_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/polynomial_ring/plot_01_division_and_gcd`

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      <div class="sphx-glr-thumbnail-title">Polynomial long division and the Euclidean algorithm</div>
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Group structure
---------------

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


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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
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  :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
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  :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>
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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
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  :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>
    </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
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  :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>
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Subgroups and cosets
----------------------

Subgroup and coset enumeration, and Lagrange's theorem.


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    <div class="sphx-glr-thumbcontainer" tooltip="Enumerates every subgroup of Z_12 and confirms each subgroup&#x27;s order divides 12, and that its left cosets partition the whole group.">

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  .. image:: /api/gallery/abstract_algebra/subgroups/images/thumb/sphx_glr_plot_01_lagranges_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/abstract_algebra/subgroups/plot_01_lagranges_theorem`

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      <div class="sphx-glr-thumbnail-title">Lagrange's theorem: subgroup order divides group order</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/abstract_algebra/actions/index.rst
   /api/gallery/abstract_algebra/codes/index.rst
   /api/gallery/abstract_algebra/finite_fields/index.rst
   /api/gallery/abstract_algebra/groups/index.rst
   /api/gallery/abstract_algebra/homomorphisms/index.rst
   /api/gallery/abstract_algebra/polynomial_ring/index.rst
   /api/gallery/abstract_algebra/structure/index.rst
   /api/gallery/abstract_algebra/subgroups/index.rst



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