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Math and science::Algebra::Aluffi

Groups. Group datum.

The datum of a group can be represented completely by a table that meets certain criteria. For a group \( G \) with set \( \{e_G, a, b, c, d, x, y, z \} \) and operation \( \cdot_G \), the table below is a scaffold for the multiplication table for \( \cdot_G \). The row and column corresponding to the identity is filled in, as the values are know from the definition of a group.

Table for \( \cdot_G \)
\( e_G \) a b c d
 x
 y z
\( e_G \) \( e_G \) 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 \( G \), and in particular, it records sufficient information to confirm that the group meets conditions such has having an identify and the presence of element inverses.

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.

\( g^n \)

\( g, g^2, g^3, g^4, ... \) can be followed by looking at just a single column (or row) of the table.

Commutativity

If a group is commutative, then the table is symmetric along the diagonal, making a large portion of the table redundant.