// Numbas version: finer_feedback_settings
{"name": "Conjugate of permutations", "extensions": ["permutations"], "custom_part_types": [{"source": {"pk": 68, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/68/edit"}, "name": "Write a permutation in cycle notation", "short_name": "write-a-permutation-in-cycle-notation", "description": "<p>The student must write a permutation in cycle notation. You can optionally require the cycles to be disjoint and/or all of length 2.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "let(ans,\n  if(settings[\"must_be_transpositions\"],\n    as_transpositions(settings[\"correct_answer\"]),\n    show(settings[\"correct_answer\"])\n  ),\n  if(ans=\"\",settings[\"identity_name\"],ans)\n)", "hint": {"static": false, "value": "if(settings[\"show_hint\"],\n  if(settings[\"must_be_disjoint\"],\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of disjoint transpositions.\",\n      \"Write a permutation in cycle notation as a product of disjoint cycles.\"\n    ),\n    if(settings[\"must_be_transpositions\"],\n      \"Write a permutation in cycle notation as a product of transpositions.\",\n      \"Write a permutation in cycle notation.\"\n    )\n  ),\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)\n\ninterpreted_answer:\nstudent_permutation\n\nis_correct:\ncorrectif(student_permutation=settings[\"correct_answer\"])\n\nis_disjoint:\nif(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)\n\nstudent_permutation:\ntry(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)\n\nis_transpositions:\nif(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(student_permutation);\napply(is_correct);\napply(is_disjoint);\napply(is_transpositions)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "student_permutation"}, {"name": "is_correct", "description": "", "definition": "correctif(student_permutation=settings[\"correct_answer\"])"}, {"name": "is_disjoint", "description": "", "definition": "if(is_disjoint(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_disjoint\"],\n    set_credit(0,\"You have not written your answer as a product of disjoint cycles.\"),\n  );\n  false\n)"}, {"name": "student_permutation", "description": "", "definition": "try(\n  if(studentAnswer=settings[\"identity_name\"],\n    permutation(\"\"), \n    permutation(studentAnswer)\n  ),\n  message,\n    warn(\"Your answer is not valid cycle notation: \"+message);\n    fail(\"Your answer is not valid cycle notation: \"+message)\n)"}, {"name": "is_transpositions", "description": "", "definition": "if(is_transpositions(studentAnswer),\n  true\n,\n  assert(not settings[\"must_be_transpositions\"],\n    set_credit(0,\"You have not written your answer as a product of transpositions.\")\n  );\n  false\n)"}], "settings": [{"name": "correct_answer", "label": "Correct answer", "help_url": "", "hint": "", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "show_hint", "label": "Show answer input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "must_be_disjoint", "label": "Must the cycles be disjoint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "must_be_transpositions", "label": "Must the cycles be transpositions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "identity_name", "label": "Name of the identity permutation", "help_url": "", "hint": "The student should write this string for the identity permutation.", "input_type": "string", "default_value": "e", "subvars": true}], "public_availability": "always", "published": true, "extensions": ["permutations"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Conjugate of permutations", "tags": ["permutations"], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "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": {}, "extensions": ["permutations"], "variables": {"sigma1": {"name": "sigma1", "group": "Ungrouped variables", "definition": "perm(randomcycle(7))", "description": "", "templateType": "anything"}, "conjugate1": {"name": "conjugate1", "group": "Ungrouped variables", "definition": "tau1*sigma1*inverse(tau1)", "description": "", "templateType": "anything"}, "tau1": {"name": "tau1", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}, "tau2": {"name": "tau2", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}, "conjugate2": {"name": "conjugate2", "group": "Ungrouped variables", "definition": "tau2*sigma2*inverse(tau2)", "description": "", "templateType": "anything"}, "sigma2": {"name": "sigma2", "group": "Ungrouped variables", "definition": "perm(deal(random(5..11)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["tau1", "tau2", "sigma1", "sigma2", "conjugate1", "conjugate2"], "variable_groups": [], "functions": {"randompartition": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "definition": "var l = [];\nwhile(n) {\n  var t = Numbas.math.randomint(n)+1;\n  l.push(t);\n  n -= t;\n}\nreturn l;"}, "tracemap": {"parameters": [["n", "number"], ["perms", "list"]], "type": "string", "language": "javascript", "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;"}, "sum": {"parameters": [["l", "list"]], "type": "number", "language": "javascript", "definition": "var t = 0;\nfor(var i=0;i<l.length;i++) {\n  t = Numbas.math.add(t,l[i]);\n}\nreturn t;"}, "permwithorder": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "randomcycle": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "primepowerfactors": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}, "showlist": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "orderpossible": {"parameters": [["n", "number"], ["s_n", "number"]], "type": "boolean", "language": "javascript", "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;"}, "permfrompartition": {"parameters": [["cycle_sizes", "list"]], "type": "list", "language": "javascript", "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;"}, "showsum": {"parameters": [["l", "list"]], "type": "string", "language": "javascript", "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;"}, "show_factorisation": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "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;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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]]</p>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "conjugate1", "must_be_disjoint": false, "must_be_transpositions": false}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>", "gaps": [{"type": "write-a-permutation-in-cycle-notation", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer": "conjugate2", "must_be_disjoint": false, "must_be_transpositions": false}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}