Groups. Group datum.
The datum of a group can be represented completely by a table that meets
certain criteria. For a group
Table for |
||||||||
---|---|---|---|---|---|---|---|---|
a | b | c | d |
x |
y | z | ||
a | b | c | d | x | y | z | ||
a | a |
|||||||
b | b | |||||||
c | c | |||||||
d | d | |||||||
x | x | |||||||
y | y | |||||||
z |
z |
Sufficiency
The table has enough information to
record all aspects of group
The reverse side lists an interesting property of the rows and columns of the table. Can you remember it? Dummy cloze
Sudoku property
An interesting aspect of a group's table is that every row and every column must contain every element once and only once, like a Sudoku puzzle. This requirement is in addition to the requirement that the table represents a function (one with two inputs). So, the group table can be considered a 3D function table with extra requirements, and then compressed down to 2D for legibility.
An interesting question is whether the sudoku puzzle property is sufficient to define a group? On first inspection, we see that this is not the case: the Sudoku property would allow two rows to exchange position, or two columns; however, doing this for the group table would break the behaviour of the identity. What if we fix the values of the operation when there is an identity input? Unfortunately, still no (I think). The Sudoku property insures that every element has a left and right inverse, but it doesn't seem like it insures these inverses are the same (not 100% sure).
Only 1 possible table for 1, 2 and 3 element groups
Interestingly, there is only one possible table for groups with 1, 2 or 3 elements. There are only 2 possible tables for a 4 element group. All tables for 1, 2, 3 and 4 element groups display commutativity of the group operation.
Commutativity
If a group is commutative, then the table is symmetric along the diagonal, making a large portion of the table redundant.