The orders of every element of the multiplicative group of remainders modulo 11.
Last active
March 16, 2017 03:52
-
-
Save bmershon/7938f064dc2202364cdd52acbd24805d to your computer and use it in GitHub Desktop.
Multiplicative Group
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| border: no | |
| license: MIT |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| <!DOCTYPE html> | |
| <meta charset="utf-8"> | |
| <script src="https://d3js.org/d3.v4.min.js"></script> | |
| <body></body> | |
| <script> | |
| let n = 11; | |
| let worker = new Worker("worker.js"); | |
| worker.onmessage = function(event) { | |
| switch (event.data.type) { | |
| case "end": return ended(event.data); | |
| } | |
| }; | |
| worker.postMessage({'n': n}); | |
| function ended(data) { | |
| let matrix = data.matrix, | |
| orders = data.orders; | |
| // Add group elements down the left side of the table. | |
| matrix = matrix.map((row, i) => { | |
| return [i + 1].concat(row).concat([orders[i]]); | |
| }); | |
| // Add group elements as top row. | |
| matrix.unshift(d3.range(0, n + 1)); | |
| // Top left corner is empty. | |
| matrix[0][0] = ""; | |
| // Top right corner is empty. | |
| matrix[0][n] = "order"; | |
| let tr = d3.select("body") | |
| .style("font-family", "helvetica") | |
| .append("table") | |
| .style("text-align", "center") | |
| .selectAll("tr") | |
| .data(matrix) | |
| .enter().append("tr"); | |
| let td = tr.selectAll("td") | |
| .data(d => d) | |
| .enter().append("td") | |
| .attr("width", "40px") | |
| .attr("height", "40px") | |
| .style("font-size", "24px") | |
| .style("font-weight", (d, i) => { | |
| switch(i) { | |
| case 0: // Left-most column. | |
| return "normal"; | |
| break; | |
| case n: // Right-most column | |
| return "bolder"; | |
| break; | |
| default: // All other entries. | |
| return "lighter"; | |
| break; | |
| } | |
| }) | |
| .style("color", (d, i) => { | |
| return (i === n) ? "rgb(240, 0, 0)" : null; | |
| }) | |
| .text(d => d); | |
| // Style the top row. | |
| d3.selectAll("tr").filter((d, i) => i === 0) | |
| .selectAll("td") | |
| .style("color", "normal"); | |
| } | |
| </script> |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| onmessage = function(event) { | |
| let n = event.data.n, | |
| matrix = multiplicative_matrix(n), | |
| orders = matrix.map((d, i) => { | |
| return order(v(i), matrix); | |
| }); | |
| postMessage({ | |
| "type": "end", | |
| "matrix": matrix, | |
| "orders": orders | |
| }); | |
| } | |
| // The value correction for a zero-based index system. | |
| function v(index) { return index + 1; } | |
| // Returns a square matrix[row][column] with n - 1 rows, | |
| // each of which is a permutation of remainders modulo n. | |
| // The first row corresponds to the element 1; the last row | |
| // corresponds to the element n - 1. | |
| function multiplicative_matrix(n) { | |
| let matrix = []; | |
| for (let i = 0; i < n - 1; i++) { | |
| matrix.push(permutation(v(i), n)); | |
| } | |
| return matrix; | |
| } | |
| // Returns an array representing a permutation of remainders modulo n. | |
| function permutation(a, n) { | |
| let row = new Array(n - 1); | |
| for (i = 0; i < n - 1; i++) { | |
| row[i] = v(i); | |
| } | |
| return row.map((d) => { | |
| return (a * d) % n; | |
| }); | |
| } | |
| // Precondition: The matrix representing the modular multiplication | |
| // table must have been constructed for a prime value n (e.g. 17). | |
| // Returns the order of the group element a, using the matrix | |
| // a to iteratively look up the result of successively applying a. | |
| function order(a, matrix) { | |
| let count = 1, | |
| n = matrix[0][matrix.length - 1] + 1, | |
| current = a; | |
| while (current !== 1) { | |
| // Zero-based index; subtract one from actual value. | |
| current = matrix[current - 1][a - 1] % n; | |
| count++; | |
| } | |
| return count; | |
| } |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment