ref: 3b83debfa8a7a5dc82c40d6500d35a9c98135cea
dir: /html/group.html/
unfinished:Group <p> Fill in the grid with the letters shown to the top and left of it, so that the full grid is a valid <a href="http://en.wikipedia.org/wiki/Cayley_table">Cayley table</a> for a <a href="http://en.wikipedia.org/wiki/Group_(mathematics)">group</a>. <p> If you don't already know what a group is, I don't really recommend trying to play this game. But if you want to try anyway, the above is equivalent to saying that the following conditions must be satisfied: <ul> <li> <strong>Latin square</strong>. Every row and column must contain exactly one of each letter. <li> <strong>Identity</strong>. There must be some letter <i>e</i> such that, for all <i>a</i>, the letter in row <i>e</i> column <i>a</i> and the one in row <i>a</i> column <i>e</i> are both <i>a</i>. In the default mode, this letter is always <i>e</i> and its row and column are filled in for you; by reconfiguring the game using the Type menu, you can select a mode in which you have to work out which letter is the identity. <li> <strong>Inverses</strong>. For every letter <i>a</i>, there must be some letter <i>b</i> (which may sometimes be the same letter as <i>a</i>) such that the letters in row <i>a</i> column <i>b</i> and in row <i>b</i> column <i>a</i> are both the identity letter (as defined above). <li> <strong>Associativity</strong>. For every combination of letters <i>a</i>, <i>b</i>, and <i>c</i>, denote the letter in row <i>a</i> column <i>b</i> by <i>d</i>, and the one in row <i>b</i> column <i>c</i> by <i>e</i>. Then the letters in row <i>d</i> column <i>c</i> and in row <i>a</i> column <i>e</i> must be the same. </ul> <p> To place a letter, click in a square to select it, then type the letter on the keyboard. To erase a letter, click to select a square and then press Backspace. <p> Right-click in a square and then type a letter to add or remove the number as a pencil mark, indicating letters that you think <em>might</em> go in that square. <p> You can rearrange the order of elements in the rows and columns by dragging the column or row headings back and forth. (The rows and columns will stay in sync with each other.) Also, left-clicking <em>between</em> two row or column headings will add or remove a thick line between those two rows and the corresponding pair of columns (which is useful if you're considering a subgroup and its cosets).