Writing group tables using formula??

Question

I am sorry for asking, but on searching for group table of $U_{24}$ came across this answer, with no hint except in a comment (by @Dylan Yott) to it that states some reference to formula for writing the multiplication table.

If given some sort of initiation into this, as am unable to google by any term as :"multiplication table of group generated by formula". Is it a sort of generating function based approach? It is my wild guess, but only that fits the formula approach for permutations.

Addendum A very good book here (Combinatorial Organic Chemistry: An Educational Approach By Sherif El-Basil), but the lack of first 14 pages make a novice like me to be unable to understand. Although, it is abound with this type only.

Is there a source available for free, that allows one to understand.

Answer 1

From the linked question:

I'm trying to show that $Z_2 \times Z_2 \times Z_2$ is isomorphic to $U(24)$.

I start by defining $f \colon Z_2 \times Z_2 \times Z_2 \to U(24)$ by $f((a,b,c)) = 12a + 6b + 4c + 1 \pmod{24}$.

The OP is trying to show that two groups are isomorphic, and uses a formula to propose what they hope is an isomorphism; a function taking elements of one group to another (with other properties).

The comment in question:

[M]aybe write out the group tables for each group and see if you can "see" an isomorphism. I wouldn't recommend going the formula route. Any chance you know anything about the classification of finite Abelian groups?

This comment is suggesting that, instead of trying to define an isomorphism by writing out a formula (as OP has attempted), they instead look at the multiplication tables (or, "Cayley tables") of the two groups in question. I assume the commenter hopes that "structural similarities" would be evident to the OP, when comparing the two multiplication tables, and that these similarities could be used in the quest for isomorphism.

So, the formula (which, presumably, doesn't work and consequently wouldn't be likely to help you understand either group) is one thing, and a multiplication table is another. The two aren't related, at least in this instance.

If you want to create a multiplication table for either group, I wouldn't expect any sort of formula to help (other than formulas defining the group operation). Note: for some groups, one may well find formulas that help in writing down a multiplication table (dihedral groups, for instance; see the last few lines in the linked section) but I wouldn't expect such a thing here.