permute-sage-exercises.ipynb

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{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h2 class=\"heading hide-type\" alt=\"Exercises 5.5 Sage Exercises\"><span class=\"type\">Section</span><span class=\"codenumber\">5.5</span><span class=\"title\">Sage Exercises</span></h2><a href=\"permute-sage-exercises.ipynb\" class=\"permalink\">¶</a></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-943\">These exercises are designed to help you become familiar with permutation groups in Sage.</p></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-224\"><h6 class=\"heading\"><span class=\"codenumber\">1</span></h6><p id=\"p-944\">Create the full symmetric group $S_{10}$ with the command <code class=\"code-inline tex2jax_ignore\">G = SymmetricGroup(10)</code>.</p></article></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-225\"><h6 class=\"heading\"><span class=\"codenumber\">2</span></h6><p id=\"p-945\">Create elements of <code class=\"code-inline tex2jax_ignore\">G</code> with the following (varying) syntax.  Pay attention to commas, quotes, brackets, parentheses.  The first two use a string (characters) as input, mimicking the way we write permuations (but with commas).  The second two use a list of tuples.</p><ul class=\"disc\"><li id=\"li-237\"><p id=\"p-946\"><code class=\"code-inline tex2jax_ignore\">a = G(\"(5,7,2,9,3,1,8)\")</code></p></li><li id=\"li-238\"><p id=\"p-947\"><code class=\"code-inline tex2jax_ignore\">b = G(\"(1,3)(4,5)\")</code></p></li><li id=\"li-239\"><p id=\"p-948\"><code class=\"code-inline tex2jax_ignore\">c = G([(1,2),(3,4)])</code></p></li><li id=\"li-240\"><p id=\"p-949\"><code class=\"code-inline tex2jax_ignore\">d = G([(1,3),(2,5,8),(4,6,7,9,10)])</code></p></li></ul><ol class=\"lower-alpha\"><li id=\"li-241\"><p id=\"p-950\">Compute $a^3\\text{,}$ $bc\\text{,}$ $ad^{-1}b\\text{.}$</p></li><li id=\"li-242\"><p id=\"p-951\">Compute the orders of each of these four individual elements (<code class=\"code-inline tex2jax_ignore\">a</code> through <code class=\"code-inline tex2jax_ignore\">d</code>) using a single permutation group element method.</p></li><li id=\"li-243\"><p id=\"p-952\">Use the permutation group element method <code class=\"code-inline tex2jax_ignore\">.sign()</code> to determine if $a,b,c,d$ are even or odd permutations.</p></li><li id=\"li-244\"><p id=\"p-953\">Create two cyclic subgroups of $G$ with the commands: </p><ul class=\"disc\"><li id=\"li-245\"><p id=\"p-954\"><code class=\"code-inline tex2jax_ignore\">H = G.subgroup([a])</code></p></li><li id=\"li-246\"><p id=\"p-955\"><code class=\"code-inline tex2jax_ignore\">K = G.subgroup([d])</code></p></li></ul><p> List, and study, the elements of each subgroup.  Without using Sage, list the order of each subgroup of $K\\text{.}$  Then use Sage to construct a subgroup of $K$ with order 10.</p></li><li id=\"li-247\"><p id=\"p-956\">More complicated subgroups can be formed by using two or more generators.  Construct a subgroup $L$ of $G$ with the command <code class=\"code-inline tex2jax_ignore\">L = G.subgroup([b,c])</code>.  Compute the order of $L$ and list all of the elements of $L\\text{.}$</p></li></ol></article></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-226\"><h6 class=\"heading\"><span class=\"codenumber\">3</span></h6><p id=\"p-957\">Construct the group of symmetries of the tetrahedron (also the alternating group on 4 symbols, $A_4$) with the command <code class=\"code-inline tex2jax_ignore\">A=AlternatingGroup(4)</code>.  Using tools such as orders of elements, and generators of subgroups, see if you can find <em class=\"emphasis\">all of</em> the subgroups of $A_4$ (each one exactly once).  Do this without using the <code class=\"code-inline tex2jax_ignore\">.subgroups()</code> method to justify the correctness of your answer (though it might be a convenient way to check your work).</p><p id=\"p-958\">Provide a nice summary as your answer—not just piles of output.  So use Sage as a tool, as needed, but basically your answer will be a concise paragraph and/or table.  This is the one part of this assignment without clear, precise directions, so spend some time on this portion to get it right.  Hint: no subgroup of $A_4$ requires more than two generators.</p></article></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-cube-symmetry\"><h6 class=\"heading\"><span class=\"codenumber\">4</span></h6><p id=\"p-959\">The subsection <a href=\"permute-sage.ipynb#subsection-motion-group-cube\" class=\"xref\" alt=\"Subsection  Motion Group of a Cube\" title=\"Subsection  Motion Group of a Cube\">Motion Group of a Cube</a> describes the $24$ symmetries of a cube as a subgroup of the symmetric group $S_8$ generated by three quarter-turns.  Answer the following questions about this symmetry group.</p><ol class=\"lower-alpha\"><li id=\"li-248\"><p id=\"p-960\">From the list of elements of the group, can you locate the ten rotations about axes?  (Hint: the identity is easy, the other nine never send any symbol to itself.)</p></li><li id=\"li-249\"><p id=\"p-961\">Can you identify the six symmetries that are a transposition of diagonals?  (Hint: <code class=\"code-inline tex2jax_ignore\">[g for g in cube if g.order() == 2]</code> is a good preliminary filter.)</p></li><li id=\"li-250\"><p id=\"p-962\">Verify that any two of the quarter-turns (<code class=\"code-inline tex2jax_ignore\">above</code>, <code class=\"code-inline tex2jax_ignore\">front</code>, <code class=\"code-inline tex2jax_ignore\">right</code>) are sufficient to generate the whole group.  How do you know each pair generates the entire group?</p></li><li id=\"li-251\"><p id=\"p-963\">Can you express one of the diagonal transpositions as a product of quarter-turns?  This can be a notoriously difficult problem, especially for software.  It is known as the “word problem.”</p></li><li id=\"li-252\"><p id=\"p-964\">Number the six faces of the cube with the numbers $1$ through $6$ (any way you like).  Now consider the same three symmetries we used before (quarter-turns about face-to-face axes), but now view them as permutations of the six faces.  In this way, we construct each symmetry as an element of $S_6\\text{.}$  Verify that the subgroup generated by these symmetries is the whole symmetry group of the cube.  Again, rather than using three generators, try using just two.</p></li></ol></article></div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-228\"><h6 class=\"heading\"><span class=\"codenumber\">5</span></h6><p id=\"p-965\">Save your work, and then see if you can crash your Sage session by building the subgroup of $S_{10}$ generated by the elements <code class=\"code-inline tex2jax_ignore\">b</code> and <code class=\"code-inline tex2jax_ignore\">d</code> of orders $2$ and $30$ from above.  <em class=\"emphasis\">Do not submit</em> the list of elements of <code class=\"code-inline tex2jax_ignore\">N</code> as part of your submitted worksheet.</p>{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["N = G.subgroup([b,d])\nN.list()"], "outputs":[]}<p id=\"p-966\">What is the order of $N\\text{?}$</p></article></div>"]}
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