// Numbas version: exam_results_page_options
{"name": "Conjugate of permutations", "extensions": ["permutations"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"orderpossible": {"definition": "var factors = Numbas.math.factorise(n);\nvar t = 0;\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    t += Math.pow(Numbas.math.primes[i],factors[i]);\n  }\n}\nreturn t<=s_n;", "type": "boolean", "language": "javascript", "parameters": [["n", "number"], ["s_n", "number"]]}, "randomcycle": {"definition": "var off = Numbas.math.randomint(n);\nvar d = Numbas.math.deal(n);\nwhile(d[off]==off) {\n  d = Numbas.math.deal(n);\n}\nvar out = [];\nfor(var i = 0;i<n;i++) {\n  out[i]=i;\n}\nvar old = off;\nvar i = off;\nwhile((i=d[i])!=off) {\n  out[old]=i;\n  old = i;\n}\nout[old] = i;\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "show_factorisation": {"definition": "var factors = Numbas.math.factorise(n);\nvar out = '';\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    if(out.length) {\n      out += ' \\\\cdot ';\n    }\n    out += Numbas.math.primes[i];\n    if(factors[i]>1) {\n      out += '^{'+factors[i]+'}';\n    }\n  }\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["n", "number"]]}, "sum": {"definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;", "type": "number", "language": "javascript", "parameters": [["l", "list"]]}, "permwithorder": {"definition": "var factors = Numbas.math.factorise(n);\nvar off = 0;\nvar out = [];\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]>0) {\n    var cycle_length = Math.pow(Numbas.math.primes[i],factors[i]);\n    for(var j=0;j<cycle_length-1;j++) {\n      out[off+j]=off+j+1;\n    }\n    out[off+cycle_length-1]=off;\n    off = off+cycle_length;\n  }\n}\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "primepowerfactors": {"definition": "var factors = Numbas.math.factorise(n);\nvar out = [];\nfor(var i=0;i<factors.length;i++) {\n  if(factors[i]) {\n    out.push(Math.pow(Numbas.math.primes[i],factors[i]));\n  }\n}\nreturn out;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "showlist": {"definition": "var out = '';\nfor(var i=0;i<l.length;i++) {\n  if(i==l.length-1 && l.length>1) {\n    out += ' and ';\n  } else if(i>0) {\n    out += ', ';\n  }\n  out += '$'+l[i]+'$';\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["l", "list"]]}, "permfrompartition": {"definition": "var to = [];\nvar offset = 0;\nfor(var i=0; i<cycle_sizes.length; i++) {\n  var o_offset = offset;\n  for(var j=0;j<cycle_sizes[i]-1;j++) {\n    to[offset] = offset+1;\n    offset += 1;\n  }\n  to[offset] = o_offset;\n  o_offset = offset;\n  offset += 1;\n}\nreturn to;", "type": "list", "language": "javascript", "parameters": [["cycle_sizes", "list"]]}, "showsum": {"definition": "var t = 0;\nvar out = '';\nfor(var i=0;i<l.length;i++) {\n  if(i>0) {\n    out += ' + ';\n  }\n  out += l[i]\n  t += l[i];\n}\nout += ' = '+t;\nreturn out;", "type": "string", "language": "javascript", "parameters": [["l", "list"]]}, "randompartition": {"definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;", "type": "list", "language": "javascript", "parameters": [["n", "number"]]}, "tracemap": {"definition": "n -= 1;\nvar l = perms.length;\nvar out = ''+(n+1);\nfor(var i=l-1;i>=0;i--) {\n  n = perms[i].map(n);\n  out += ' \\\\to '+(n+1);\n}\nreturn out;", "type": "string", "language": "javascript", "parameters": [["n", "number"], ["perms", "list"]]}}, "name": "Conjugate of permutations", "tags": ["permutations"], "type": "question", "advice": "<p>You could find $\\tau \\sigma \\tau^{-1}$ by first finding $\\tau^{-1}$ and then performing the composition by hand, but there's a quicker way: $\\tau \\sigma \\tau^{-1}$ has the same cycle structure as $\\sigma$, but each $n$ is replaced by $\\tau(n)$.</p>\n<h5>Proof</h5>\n<p>Assume, without loss of generality, that $\\sigma$ is a single cycle $(b_1, \\dots, b_n)$, and let $\\pi = \\tau \\sigma \\tau^{-1}$. Look at what happens to $\\tau(b_i)$ when acted on by $\\sigma$:</p>\n<p>\\[ \\tau(b_i) \\overset{\\tau^{-1}}\\mapsto b_i \\overset{\\sigma}\\mapsto b_{i+1} \\overset{\\tau}\\mapsto \\tau(b_{i+1}) \\]</p>\n<p>So, for any $1 \\lt i \\leq n$, $\\tau(b_i) \\overset{\\pi}\\mapsto \\tau(b_{i+1})$.</p>\n<p>For any $x \\notin \\{b_1,\\dots,b_n\\}$, $\\sigma(x) = x$, so $\\pi(x) = x$.</p>\n<p>Hence, $\\pi$ is the cycle $(\\tau(b_1), \\dots, \\tau(b_n))$.</p>\n<h4>a)</h4>\n<p>$\\var{conjugate1}$</p>\n<h4>b)</h4>\n<p>$\\var{conjugate2}$</p>\n", "rulesets": {}, "parts": [{"prompt": "<p>Where $\\sigma = \\var{sigma1}$ and $\\tau = \\var{tau1}$, write down the permutation $\\tau \\sigma \\tau^{-1}$.</p>\n<p>$\\tau \\sigma \\tau^{-1}$ = [[0]]\n</p>", "marks": 0, "gaps": [{"marks": 1, "showCorrectAnswer": true, "displayAnswer": "{show(conjugate1)}", "scripts": {"validate": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry{\n  Permutation.parse(this.studentAnswer);\n} catch(e) {\n  this.giveWarning(\"Invalid notation\");\n  return false;\n}\nreturn true;"}, "mark": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry {\n  var p = Permutation.fromString(this.studentAnswer);\n} catch(e) {\n  this.setCredit(0,\"Invalid notation\");\n  return;\n}\nif(!Permutation.is_disjoint(this.studentAnswer)) {\n  this.setCredit(0,\"Your cycles are not disjoint\");\n  return;\n}\nvar unwrapValue = Numbas.jme.unwrapValue;\nvar prod = unwrapValue(question.scope.variables.conjugate1);\nif(p.equal(prod)) {\n  this.setCredit(1,\"Correct!\");\n} else {\n  this.setCredit(0,\"Wrong\");\n}"}}, "answer": "", "type": "patternmatch"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Where $\\sigma = \\var{sigma2}$ and $\\tau = \\var{tau2}$, write down the permutation $\\tau \\sigma \\tau^{-1}$.</p>\n<p>$\\tau \\sigma \\tau^{-1}$ = [[0]]\n</p>", "marks": 0, "gaps": [{"marks": 1, "showCorrectAnswer": true, "displayAnswer": "{show(conjugate2)}", "scripts": {"validate": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry{\n  Permutation.parse(this.studentAnswer);\n} catch(e) {\n  this.giveWarning(\"Invalid notation\");\n  return false;\n}\nreturn true;"}, "mark": {"order": "instead", "script": "var Permutation = Numbas.extensions.permutations.Permutation;\ntry {\n  var p = Permutation.fromString(this.studentAnswer);\n} catch(e) {\n  this.setCredit(0,\"Invalid notation\");\n  return;\n}\nif(!Permutation.is_disjoint(this.studentAnswer)) {\n  this.setCredit(0,\"Your cycles are not disjoint\");\n  return;\n}\nvar unwrapValue = Numbas.jme.unwrapValue;\nvar prod = unwrapValue(question.scope.variables.conjugate2);\nif(p.equal(prod)) {\n  this.setCredit(1,\"Correct!\");\n} else {\n  this.setCredit(0,\"Wrong\");\n}"}}, "answer": "", "type": "patternmatch"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "", "variable_groups": [], "progress": "in-progress", "preamble": {"css": "", "js": ""}, "variables": {"conjugate1": {"definition": "tau1*sigma1*inverse(tau1)", "templateType": "anything", "group": "Ungrouped variables", "name": "conjugate1", "description": ""}, "conjugate2": {"definition": "tau2*sigma2*inverse(tau2)", "templateType": "anything", "group": "Ungrouped variables", "name": "conjugate2", "description": ""}, "tau2": {"definition": "perm(deal(random(5..11)))", "templateType": "anything", "group": "Ungrouped variables", "name": "tau2", "description": ""}, "tau1": {"definition": "perm(deal(random(5..11)))", "templateType": "anything", "group": "Ungrouped variables", "name": "tau1", "description": ""}, "sigma1": {"definition": "perm(randomcycle(7))", "templateType": "anything", "group": "Ungrouped variables", "name": "sigma1", "description": ""}, "sigma2": {"definition": "perm(deal(random(5..11)))", "templateType": "anything", "group": "Ungrouped variables", "name": "sigma2", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}