Fill the grid with the letters along its top and left. With row a, column b read as a·b:
On a touch screen, a tap does what a click does, and a long press what a right-click does (more ways to right-click). A key the puzzle’s controls show as a button can be pressed there instead of on a keyboard.
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.
Right-click in a square and then type a letter to add or remove the letter as a pencil mark, indicating letters that you think might go in that square.
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 between 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).
You can also use the arrow keys to move the selected square around. Press Enter to toggle between entering letters and entering pencil marks, and Backspace or Space to clear a square.
Press Fill marks among the puzzle’s controls to write in every pencil mark wherever nothing is placed and no mark is written yet. The M key does the same, on any board where M is not something you type. Marks you have narrowed are left alone, and pressing again never starts over: see Filling in all the pencil marks. In Group, M is a letter you type on a grid of 13 letters or more, so there only the button fills the marks. Once every empty square has marks, the press crosses out of each the letters already placed in its row and column.
From the keyboard, Shift and Left or Right moves the selected square’s column one place along the order, and Shift and Up or Down moves its row; the rows and columns stay in sync either way. Press | to add or remove the thick line after the selected column, and - for the line below the selected row.
A grid that meets them is the Cayley table of a group, but the game needs no more of the mathematics than this:
Hint explains the next step rather than simply making it. It works from your own letters and pencil marks, so it carries on from wherever you are; if a letter you’ve placed is wrong, or a cell’s pencil marks have crossed out its answer, it asks you to fix the highlighted mistakes first. It pencils in only as it needs to: a cell with no marks counts as holding every element its row and column don’t already hold, and the hint writes a cell’s marks only when its next step crosses one out or reasons from them. If you would rather it filled in every cell’s marks first (the same as pressing Fill marks), set Hints pencil in to Every candidate first in the preferences.
The hint writes products the way the grid shows them: a·b is the letter in row a, column b. The step that belongs to Group alone is associativity: once the grid shows a·b, b·c and one of (a·b)·c and a·(b·c), the other must be the same letter. When you have to find the identity yourself, a product like a·b = a shows that b is the identity, and a product that changes a letter rules its partner out as the identity; the hint then crosses out that element’s identity marks, the pencil mark in each cell of its row and column that would have made it the identity there.
On the hardest boards that need no guessing, a step may be a wing. Take a cell whose pencil marks are down to two values, say a or b, and two cells it can’t repeat a value with, both outlined: one marked a and c, the other b and c. Whatever goes in the middle cell, one of the outlined cells must have c, so a cell that can’t repeat a value with either of them can’t be c. If the middle cell could be c as well, the same holds for a cell that can’t repeat a value with the middle cell either.
Rearranging the rows and columns doesn’t disturb a hint: its marks follow the elements to wherever you’ve dragged them. On an Unreasonable board, deduction can run out before the grid is full; the hint says so rather than guessing.
These parameters are available from the ‘Custom type…’ option on the ‘Type’ menu.