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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 4.8 Sage Exercises\"><span class=\"type\">Section</span><span class=\"codenumber\">4.8</span><span class=\"title\">Sage Exercises</span></h2><a href=\"cyclic-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-773\">This group of exercises is about the group of units mod $n\\text{,}$ $U(n)\\text{,}$ which is sometimes cyclic, sometimes not. There are some commands in Sage that will answer some of these questions very quickly, but instead of using those now, just use the basic techniques described. The idea here is to just work with elements, and lists of elements, to discern the subgroup structure of these groups.</p></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-774\">Sage worksheets have extensive capabilities for making new cells with carefully formatted text, include support for <span class=\"latex-logo\">L<span class=\"A\">a</span>T<span class=\"E\">e</span>X</span> syntax to express mathematics. So when a question asks for explanation or commentary, make a new cell and communicate clearly with your audience. Continue this practice in subsequent exercise sets.</p></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-183\"><h6 class=\"heading\"><span class=\"codenumber\">1</span></h6><p id=\"p-775\">Execute the statement <code class=\"code-inline tex2jax_ignore\">R = Integers(40)</code> to create the set <code class=\"code-inline tex2jax_ignore\">[0,1,2,...,39]</code> This is a group under addition mod $40\\text{,}$ which we will ignore. Instead we are interested in the subset of elements which have an inverse under <em class=\"emphasis\">multiplication</em> mod $40\\text{.}$ Determine how big this subgroup is by executing the command <code class=\"code-inline tex2jax_ignore\">R.unit_group_order()</code>, and then obtain a list of these elements with <code class=\"code-inline tex2jax_ignore\">R.list_of_elements_of_multiplicative_group()</code>.</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-184\"><h6 class=\"heading\"><span class=\"codenumber\">2</span></h6><p id=\"p-776\">You can create elements of this group by coercing regular integers into <code class=\"code-inline tex2jax_ignore\">U</code>, such as with the statement <code class=\"code-inline tex2jax_ignore\">a = U(7)</code>. (Don't confuse this with our mathematical notation $U(40)\\text{.}$) This will tell Sage that you want to view $7$ as an element of $U\\text{,}$ subject to the corresponding operations. Determine the elements of the cyclic subgroup of $U$ generated by $7$ with a list comprehension as follows:</p>{"cell_type": "code", "execution_count":null, "metadata":{}, "source":["R = Integers(40)\na = R(7)\n[a^i for i in srange(16)]"], "outputs":[]}<p id=\"p-777\">What is the order of $7$ in $U(40)\\text{?}$</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-185\"><h6 class=\"heading\"><span class=\"codenumber\">3</span></h6><p id=\"p-778\">The group $U(49)$ is cyclic. Using only the Sage commands described previously, use Sage to find a generator for this group. Now using <em class=\"emphasis\">only</em> theorems about the structure of cyclic groups, describe each of the subgroups of $U(49)$ by specifying its order and by giving an explicit generator. Do not repeat any of the subgroups — in other words, present each subgroup <em class=\"emphasis\">exactly</em> once. You can use Sage to check your work on the subgroups, but your answer about the subgroups should rely only on theorems and be a nicely written paragraph with a table, etc.</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-186\"><h6 class=\"heading\"><span class=\"codenumber\">4</span></h6><p id=\"p-779\">The group $U(35)$ is not cyclic. Again, using only the Sage commands described previously, use computations to provide irrefutable evidence of this. How many of the $16$ different subgroups of $U(35)$ can you list?</p></article></div>"]}, {"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><article class=\"exercise-like\" id=\"exercise-187\"><h6 class=\"heading\"><span class=\"codenumber\">5</span></h6><p id=\"p-780\">Again, using only the Sage commands described previously, explore the structure of $U(n)$ for various values of $n$ and see if you can formulate an interesting conjecture about some basic property of this group. (Yes, this is a <em class=\"emphasis\">very</em> open-ended question, but this is ultimately the real power of exploring mathematics with Sage.)</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": "cyclic-sage-exercises.ipynb"} }