Suppose someone (person B) knows a finite group $G$ of order $n$. You (person A) know only the order $n$, and that $1$ is the name of the identity element. The group elements are named $1,2,\ldots,n$ in arbitrary order, arbitrary except that $1$ is the identity. You are permitted to make queries of this form to B, who answers truthfully:
Tell me which element $c$ of $G$ is the product (group operation) of elements $a$ and $b$, i.e., $a \circ b =\;$?
- Q. How many queries always suffice (and are sometimes needed) to determine $G$?
This has likely been studied but I must not be using the accepted terminology in my searches.
As a simple example, how many queries are needed to distinguish between the two groups of order $4$, $K_4$ and $Z_4$? I believe that two queries suffice:
- $2 \circ 2 =\;$?
- $3 \circ 3 =\;$?
This is because in the Klein group, $a^2 = 1$ for all $a$, but in $Z_4$, only one of the three non-identity elements has a square of $1$. The first query could (unluckily) yield $2 \circ 2 = 1$, but then the second query settles it.
Added. By "determine $G$," I mean identify which of the $k$ non-isomorphic groups of order $n$ is $G$. For example, there are $5$ groups of order-$8$; there are $14$ groups of order-$16$. The goal of the queries would be to pinpoint which of these groups is $G$, up to group isomorphism.