Group table and Sudoku puzzles

For any finite group, we will notice a striking similarity between its group multiplication table and the Sudoku puzzles. Every nXn Sudoku puzzle should satisfy three rules: Every row should contain exactly those n numbers 1 through n; Every column should contain exactly those n numbers 1 throught n; In addition, if n=kXk is a perfect square, then every kXk (non-overlapping) grid should contain exactly those n numbers 1 through n. By the cancellation law of the group, every group multiplication table will automatically satisfy the first two rules. Unfortunately, it will almost always fail the last rule. One way to fix it is to allow row/column switching for the group multiplication table. A natural question is: Can all Sudoku puzzles be induced by a group in this way? The answer is: It depends. We will explore this question from both algebraic and statistical perspectives and search through computer programming to see the percentage of group-induced Sudokus among all Sudokus.