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{ "cells": [ {"cell_type": "code", "execution_count":null, "metadata":{}, "source":["%%html\n<link href=\"http://mathbook.pugetsound.edu/beta/mathbook-content.css\" rel=\"stylesheet\" type=\"text/css\" />\n<link href=\"https://aimath.org/mathbook/mathbook-add-on.css\" rel=\"stylesheet\" type=\"text/css\" />\n<style>.subtitle {font-size:medium; display:block}</style>\n<link href=\"https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic\" rel=\"stylesheet\" type=\"text/css\" />\n<link href=\"https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext\" rel=\"stylesheet\" type=\"text/css\" /><!-- Hide this cell. -->\n<script>\nvar cell = $(\".container .cell\").eq(0), ia = cell.find(\".input_area\")\nif (cell.find(\".toggle-button\").length == 0) {\nia.after(\n $('<button class=\"toggle-button\">Toggle hidden code</button>').click(\n function (){ ia.toggle() }\n )\n )\nia.hide()\n}\n</script>\n"], "outputs":[]}, {"cell_type":"markdown", "metadata":{}, "source":["**Important:** to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection. It should already be selected, or place your cursor anywhere above to select. Then press the \"Run\" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard."]}, {"cell_type":"markdown", "metadata":{}, "source":["$\\newcommand{\\identity}{\\mathrm{id}}\n\\newcommand{\\notdivide}{\\nmid}\n\\newcommand{\\notsubset}{\\not\\subset}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\gf}{\\operatorname{GF}}\n\\newcommand{\\inn}{\\operatorname{Inn}}\n\\newcommand{\\aut}{\\operatorname{Aut}}\n\\newcommand{\\Hom}{\\operatorname{Hom}}\n\\newcommand{\\cis}{\\operatorname{cis}}\n\\newcommand{\\chr}{\\operatorname{char}}\n\\newcommand{\\Null}{\\operatorname{Null}}\n\\newcommand{\\lt}{<}\n\\newcommand{\\gt}{>}\n\\newcommand{\\amp}{&}\n$"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h2 class=\"heading hide-type\" alt=\"Exercises 15.7 Sage Exercises\"><span class=\"type\">Section</span><span class=\"codenumber\">15.7</span><span class=\"title\">Sage Exercises</span></h2><a href=\"sylow-sage-exercises.ipynb\" class=\"permalink\">¶</a></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-528\"><h6 class=\"heading\"><span class=\"codenumber\">1</span></h6><p id=\"p-2294\">This exercise verifies Theorem <a href=\"section-sylow-applications.ipynb#theorem-commutator-subgroup-theorem\" class=\"xref\" alt=\"Theorem 15.13 \" title=\"Theorem 15.13 \">15.13</a>. The commutator subgroup is computed with the permutation group method <code class=\"code-inline tex2jax_ignore\">.commutator()</code>. For the dihedral group of order $40\\text{,}$ $D_{20}$ (<code class=\"code-inline tex2jax_ignore\">DihedralGroup(20)</code> in Sage), compute the commutator subgroup and form the quotient with the dihedral group. Then verify that this quotient is abelian. Can you identify the quotient group exactly (in other words, up to isomorphism)?</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-529\"><h6 class=\"heading\"><span class=\"codenumber\">2</span></h6><p id=\"p-2295\">For each possible prime, find all of the distinct Sylow $p$-subgroups of the alternating group $A_5\\text{.}$ Confirm that your results are consistent with the Third Sylow Theorem for each prime. We know that $A_5$ is a simple group. Explain how this would explain or predict some aspects of your answers.</p><p id=\"p-2296\">Count the number of distinct elements contained in the union of all the Sylow subgroups you just found. What is interesting about this count?</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-530\"><h6 class=\"heading\"><span class=\"codenumber\">3</span></h6><p id=\"p-2297\">For the dihedral group $D_{36}$ (symmetries of a $36$-gon) and each possible prime, determine the possibilities for the number of distinct Sylow $p$-subgroups as predicted by the Third Sylow Theorem (<a href=\"section-sylow-theorems.ipynb#theorem-third-sylow\" class=\"xref\" alt=\"Theorem 15.8 Third Sylow Theorem\" title=\"Theorem 15.8 Third Sylow Theorem\">15.8</a>). Now compute the actual number of distinct Sylow $p$-subgroups for each prime and comment on the result.</p><p id=\"p-2298\">It can be proved that <em class=\"emphasis\">any group</em> with order $72$ is not a simple group, using techniques such as those used in the later examples in this chapter. Discuss this result in the context of your computations with Sage.</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-531\"><h6 class=\"heading\"><span class=\"codenumber\">4</span></h6><p id=\"p-2299\">This exercise verifies Lemma <a href=\"section-sylow-theorems.ipynb#lemma-distinct-conjugates\" class=\"xref\" alt=\"Lemma 15.6 \" title=\"Lemma 15.6 \">15.6</a>. Let $G$ be the dihedral group of order $36\\text{,}$ $D_{18}\\text{.}$ Let $H$ be the one Sylow $3$-subgroup. Let $K$ be the subgroup of order $6$ generated by the two permutations <code class=\"code-inline tex2jax_ignore\">a</code> and <code class=\"code-inline tex2jax_ignore\">b</code> given below. First, form a list of the distinct conjugates of $K$ by the elements of $H\\text{,}$ and determine the number of subgroups in this list. Compare this with the index given in the statement of the lemma, employing a single (long) statement making use of the <code class=\"code-inline tex2jax_ignore\">.order()</code>, <code class=\"code-inline tex2jax_ignore\">.normalizer()</code> and <code class=\"code-inline tex2jax_ignore\">.intersection()</code> methods with <code class=\"code-inline tex2jax_ignore\">G</code>, $H$ and $K\\text{,}$ <em class=\"emphasis\">only</em>.</p>{"cell_type": "code", "execution_count":null, "metadata":{}, "source":["G = DihedralGroup(18)\na = G(\"(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)\")\nb = G(\"(1,5)(2,4)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)\")"], "outputs":[]}</article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-532\"><h6 class=\"heading\"><span class=\"codenumber\">5</span></h6><p id=\"p-2300\">Example <a href=\"section-sylow-applications.ipynb#example-group-48\" class=\"xref\" alt=\"Example 15.19 \" title=\"Example 15.19 \">15.19</a> shows that every group of order $48$ has a normal subgroup. The dicyclic groups are an infinite family of non-abelian groups with order $4n\\text{,}$ which includes the quaternions (the case of $n=2$). So the permutation group <code class=\"code-inline tex2jax_ignore\">DiCyclicGroup(12)</code> has order 48. Use Sage to follow the logic of the proof in Example <a href=\"section-sylow-applications.ipynb#example-group-48\" class=\"xref\" alt=\"Example 15.19 \" title=\"Example 15.19 \">15.19</a> and construct a normal subgroup in this group. (In other words, do not just ask for a list of the normal subgroups from Sage, but instead trace through the implications in the example to arrive at a normal subgroup, and then check your answer.)</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-533\"><h6 class=\"heading\"><span class=\"codenumber\">6</span></h6><p id=\"p-2301\">The proofs of the Second and Third Sylow Theorems (<a href=\"section-sylow-theorems.ipynb#theorem-second-sylow\" class=\"xref\" alt=\"Theorem 15.7 Second Sylow Theorem\" title=\"Theorem 15.7 Second Sylow Theorem\">15.7</a>, <a href=\"section-sylow-theorems.ipynb#theorem-third-sylow\" class=\"xref\" alt=\"Theorem 15.8 Third Sylow Theorem\" title=\"Theorem 15.8 Third Sylow Theorem\">15.8</a>) employ a group action on sets of Sylow $p$-subgroups, For the Second Theorem, the list is proposed as incomplete and is proved to be <em class=\"emphasis\">all</em> of the Sylow $p$-subgroups. In this exercise we will see how these actions behave, and how they are different when we use different groups acting on the same set.</p><p id=\"p-2302\">Construct the six Sylow $5$-subgroups of the alternating group $A_5\\text{.}$ This will be the set of objects for both of our actions. Conjugating one of these Sylow $5$-subgroups by an element of $A_5$ will produce another Sylow $5$-subgroup, and so can be used to create a group action. For such an action, from each group element form a Sage permutation of the subgroups by numbering the six subgroups and using these integers as markers for the subgroups. You will find the Python list method <code class=\"code-inline tex2jax_ignore\">.index()</code> very helpful. Now use all of these permutations to generate a permutation group (a subgroup of $S_6$). Finally, use permutation group methods for orbits and stabilisers, etc. to explore the actions.</p><p id=\"p-2303\">For the first action, use all of $A_5$ as the group. Show that the resulting action is transitive. In other words, there is exactly one single orbit.</p><p id=\"p-2304\">For the second action, use just one of the Sylow $5$-subgroups as the group. Write the class equation for this action in a format that suggests the “congruent to $1$ mod $p$” part of the conclusion of the Third Theorem.</p></article></div>"]} ], "nbformat": 4, "nbformat_minor": 0, "metadata": {"kernelspec": {"display_name": "", "name": "sagemath"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 2}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.8"}, "name": "sylow-sage-exercises.ipynb"} }