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<section>
  <h1 class="title">6G6Z3012 Group Theory</h1>
  <h2 class="author">Options talk by Dr Killian O'Brien</h2>
  <h3 class="date">MMU, February 2017</h3>
</section>

<section id="the-unit" class="slide level1">
<h1>The unit</h1>
<ul>
<li>Taught by Dr Killian O'Brien + ...</li>
<li>2 hours lecture + 1 hour tutorial/lab per week</li>
<li>Coursework problems (40%), Examination (60%)</li>
<li>New(ish) unit: introduced in response to student requests for more pure mathematics content</li>
</ul>
</section>
<section id="nature-of-the-unit" class="slide level1">
<h1>Nature of the unit</h1>
<ul>
<li>A thorough introduction to a substantial area of pure mathematics that has strong connections to areas of geometry, combinatorics, graph theory, ... .</li>
<li>Definitely suited to students who like problem solving and the unit will develop your skills in this area.</li>
<li>We will use the SageMath mathematics system to aid our investigations. You will also get an introduction to the Python programming language. (<a href="http://www.sagemath.org">www.sagemath.org</a>, <a href="https://cloud.sagemath.org">cloud.sagemath.org</a>, <a href="https://sagecell.sagemath.org">sagecell.sagemath.org</a>)</li>
    <li>Excellent textbook: <a href='http://abstract.pugetsound.edu/'><em>Abstract Algebra</em></a> by Tom Judson</li>
</ul>
</section>
<section id="what-you-will-get-from-the-unit" class="slide level1">
<h1>What you will get from the unit</h1>
<ul>
<li>An appreciation for the many aspects of group theory and its connections to other areas of mathematics.</li>
<li>Development of your problem solving and abstract thinking skills.</li>
<li>Exposure to SageMath mathematical software, aimed at pure mathematics teaching and research.</li>
<li>SageMath uses the Python programming language which is widely used in many areas of computing.</li>
</ul>
</section>
<section id="what-is-a-group" class="slide level1">
<h1>What is a group?</h1>
<p>A set of mathematical objects with a mathematically meaningful operation applied amongst them, as long as it is <em>well behaved</em>, it will be a group. It's a very broad concept and present in many areas of mathematics.</p>
<h2 id="more-formally">More formally</h2>
<p>A set <span class="math inline">\(G\)</span> with a binary operation <span class="math inline">\(\star\)</span> satisfying</p>
<ul>
<li><span class="math inline">\(G\)</span> is a closed system under <span class="math inline">\(\star\)</span>, i.e. <span class="math inline">\(x \star y \in G\)</span>.</li>
<li><span class="math inline">\(\star\)</span> is associative on <span class="math inline">\(G\)</span>, i.e. <span class="math inline">\(x \star ( y \star z) = (x \star y) \star z\)</span>.</li>
<li><span class="math inline">\(G\)</span> contains an identity element, <span class="math inline">\(e\)</span>, for <span class="math inline">\(\star\)</span>, i.e. <span class="math inline">\(x \star e = e \star x = x\)</span>.</li>
<li><span class="math inline">\(G\)</span> contains inverse elements for <span class="math inline">\(\star\)</span>, i.e. <span class="math inline">\(x \star x^{-1} = e\)</span>.</li>
</ul>
</section>
<section id="syllabus-topics" class="slide level1">
<h1>Syllabus topics</h1>
<h2 id="examples-of-groups">Examples of groups:</h2>
<ul>
<li>Symmetry groups of two and three-dimensional objects,</li>
<li>the dihedral groups <span class="math inline">\(D_n\)</span>,</li>
<li>rotational symmetry groups of three-dimensional polyhedra.</li>
<li>Permutation groups, the Symmetric groups <span class="math inline">\(S_n\)</span> and the Alternating groups <span class="math inline">\(A_n\)</span>.</li>
<li>Number based groups under arithmetic operations,</li>
<li>the cyclic groups <span class="math inline">\(\mathbb{Z}_n\)</span>.</li>
<li>Groups of matrices.</li>
<li>...</li>
</ul>
</section>
<section id="visualising-the-elements-of-gammat" class="slide level1">
<h1>Visualising the elements of <span class="math inline">\(\Gamma^+(T)\)</span></h1>
<p><span class="math inline">\(\Gamma^+(T)\)</span> is the set of rotational symmetries of the tetrahedron.</p>
<p><img height="400" src=./tetrahedrons.png></p>
</section>
<section id="the-symmetric-group-s_4-on-four-objects" class="slide level1">
<h1>The Symmetric group, <span class="math inline">\(S_4\)</span>, on four objects</h1>
<p><span class="math inline">\(S_4\)</span> is the group of all rearrangments, or <em>permutations</em>, of four objects.</p>
<p><img height="400" src=./S4_Cayley.png></p>
</section>
<section id="syllabus-topics-1" class="slide level1">
<h1>Syllabus topics</h1>
<h2 id="initial-group-theory">Initial group theory:</h2>
<p>Various concept definitions and examples, including: elements, orders, Abelian groups, subgroups, generators and cyclic/non-cyclic. The natural mappings between groups, homomorphisms and isomorphisms. Examples of isomorphic pairs and non-isomorphic pairs. Cayley's theorem: Every group isomorphic to a group of permutations.</p>
</section>
<section id="syllabus-topics-2" class="slide level1">
<h1>Syllabus topics</h1>
<h2 id="classification-problems">Classification problems:</h2>
<p>What are the grand enterprises of group theory? What classification problems can be posed?</p>
<h2 id="lagranges-theorem">Lagrange's theorem:</h2>
<p>Restricting the possibilities for subgroup orders. Equivalence relations, equivalence classes, cosets. Normal groups and quotient groups.</p>
</section>
<section id="lagranges-theorem-1" class="slide level1">
<h1>Lagrange's theorem</h1>
<p><img height="400" src=./MAA.png></p>
<p>A page from <em>Groups and Symmetry</em> by M. A. Armstrong.</p>
</section>
<section id="syllabus-topics-3" class="slide level1">
<h1>Syllabus topics</h1>
<h2 id="group-presentations">Group presentations:</h2>
<p>How to systematically describe groups in a computable way. Group presentations, generators and relations, presentation matrices. The isomorphism decision problem based on matrices.</p>
<h2 id="the-classification-of-finitely-presented-abelian-groups">The classification of finitely presented Abelian groups:</h2>
<p>A matrix reduction algorithm to decide the isomorphism problem amongst finitely presented Abelian groups. The canonical form of finitely presented Abelian group as a direct sum of cyclic groups.</p>
        </section>
<section id="syllabus-topics-3" class="slide level1">
<h1>Syllabus topics</h1>
<h2 id="classification-of-groups-of-low-order">Classification of groups of low order:</h2>
<p>What about non-Abelian groups? Why we can't solve using matrix reduction? Investigation of groups of low order and enumeration and classification of all groups up to some suitable order.</p>
<h2 id="sylows-theorems">Sylow's theorems:</h2>
<p>Discussion of the converse to Lagrange's theorem. Group actions, orbits, stabilizers. Self-action by conjugation. Sylow's theorems.</p>
</section>
<section id="wider-interest-material-applications" class="slide level1">
<h1>Wider interest material / applications</h1>
<p>The unit could contain interesting general material on the following topics/applications.</p>
<h2 id="the-classification-of-finite-simple-groups">The classification of finite simple groups</h2>
<p>The grand project. Status of the proof. Some history and biographical details of the completion of the project. The families in the classification. The sporadic groups. The Monster group and Monstrous Moonshine.</p>
</section>
<section id="wider-interest-material-applications-1" class="slide level1">
<h1>Wider interest material / applications</h1>
<h2 id="the-monster-group">The Monster group</h2>
<ul>
<li>A group, <span class="math inline">\(M\)</span>, with approx. <span class="math inline">\(8 \times 10^{53}\)</span> elements, that is <em>simple</em>, i.e. it has no <em>normal</em> subgroups.</li>
<li><span class="math inline">\(M\)</span> is (isomorphic to) a group of rotations of 196883-dimensional space.</li>
<li><span class="math inline">\(M\)</span> is (isomorphic to) a group of matrices generated by two particular binary <span class="math inline">\(196882 \times 196882\)</span> matrices.</li>
</ul>
</section>
<section id="wider-interst-material-applications" class="slide level1">
<h1>Wider interst material / applications</h1>
<h2 id="algorithmic-problems">Algorithmic problems</h2>
<p>The word problem. Computability. Alan Turing.</p>
<p><img height="400" src=./turing.jpg></p>
</section>
<section id="wider-interst-material-applications-1" class="slide level1">
<h1>Wider interst material / applications</h1>
<h2 id="combinatorial-enumeration-and-geometric-classification-problems">Combinatorial enumeration and geometric classification problems</h2>
<p>Counting number of distinguishable colourings of geometric objects. Classifying the symmetry types of two-dimensional wallpaper patterns. Classifying two and three-dimensional crystal structures (lattices).</p>
</section>
<section id="the-are-exactly-17-different-types-of-wallpaper-up-to-symmetry." class="slide level1">
<h1>The are exactly 17 different types of wallpaper, up to symmetry.</h1>
<p><img height="400" src=./wallpapers.png></p>
</section>
<section id="wider-interst-material-applications-2" class="slide level1">
<h1>Wider interst material / applications</h1>
<h2 id="galois-theory">Galois theory</h2>
<p><img height="200" src=./Galois.jpeg></p>
<ul>
<li>Life of Galois (1811 - 1832).</li>
<li>Galois theory.</li>
<li>Formulas for roots of polynomials.</li>
<li>Construction problems with ruler and compass.</li>
</ul>
</section>
<section id="sagemath" class="slide level1">
<h1>SageMath</h1>
<pre><code data-trim data-noescape>
G=DihedralGroup(12)
# G is the group of rotational and
# reflectional symmetries of the
# regular 12-sided polygon

HH = G.subgroups()
# HH is the set of all subgroups of G

# The following for loop prints the
# Cayley graph of each subgroup of G
for H in HH:
    show(H.cayley_graph())
</code></pre>
<p>The above code displays the <em>Cayley Graph</em> of every subgroup of <span class="math inline">\(D_{12}\)</span>. See the <a href="https://sagecell.sagemath.org/?z=eJw9j0GKwzAMRfeB3EHQTbqYQrssdNUB5wDdF02ixAbVMrZDye1HSjPjlfm897_tbt_B05iRXZYldefLsW0O4CAUqJ5gthRkgiwVa5CIDBhHYzJNTMOelfX1opoDFYPV_BDzwpjhfPkqYaQRkvA6S2ybtul7uIE7leVnmyjdtqvpPlyoWhOydv8xFjiTD_BQZBJmeYc46y0DiyRIOcT6KTDqjivTqp_A5E0mHPx_3d5mrq5G3b62DegpXt5dfxo2-bnJ3VGf9wsk8l-j&amp;lang=sage">results</a> from SageMath.</p>
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