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Given an $n\times n$ grid partially filled with the numbers $0,\ldots, n-1$, we can play a Sudoku-like game by trying to fill in the rest of the grid so that the end result is the Cayley table for a group (edit: where position $i,j$ in the table would represent the product of $i $ and $j$ in the group structure).

I am curious about the following value: let $g(n)$ represent the smallest natural number such that if $g(n)$ squares in an $n\times n$ grid are filled, then the configuration always can be extended to a solution in at most $1$ way.

Another way to think of this question is as follows: How similar can two distinct $n\times n$ Cayley tables be? $g(n)$ is the smallest natural number such that if two $n\times n$ Cayley tables agree for $g(n)$ entries, then they are identical.

What does $g(n)$ look like? I'm most interested in the asymptotic behaviour (as I imagine $g(n)$ may fluctuate wildly as $n$ increases due to the fluctuation in the number of groups of order $n$ as $n$ increases). One particularly interesting question: can $g(n)\over n^2$ be bounded above by a constant (say, 0.99)? In other words, if two Cayley tables agree on $99\%$ of entries, must they be identical? And if so, how much lower than $0.99$ can we bring this constant?

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    $\begingroup$ If you are asking for equality of tables (as opposed to isomorphism), probably not. For Latin Squares, take an $n\times n$ and swap the $1$s and $2$s entries. That changes $2n$ entries, and $(n^2-2n)/n^2\to 1$ as $n\to\infty$. But that leads to equivalent Latin Squares. You should probably be asking for equivalent Cayley tables or isomorphic groups, rather than identical tables. $\endgroup$ Feb 16 at 15:37
  • $\begingroup$ @ArturoMagidin I see what you are saying. I was imagining that the entry in position $i,j$ of the table would represent the product of $i$ and $j$, hence permutations of the rows/columns wouldn't be an issue, though I realize the way I wrote it left that open. I made an edit, hopefully that cleared it up. $\endgroup$
    – volcanrb
    Feb 16 at 19:33
  • $\begingroup$ This function is indeed fluctuating wildly. If $n$ happens to be a prime, there is exactly one group of that order, so even a completely empty Cayley table determines the group uniquely. For a few cases like $n=p^2$ or $n=p^3$ one can probably explicitly compute $g(n)$. For $n$ a power of $2$ gives the largest number of non-isomorphic groups and probably the biggest overlap in Cayley tables of different groups. $\endgroup$
    – quarague
    Feb 17 at 19:59

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As a partial answer to your second question: in 1992 Ales Drápal proved that if two finite groups agree on 89% on their multiplication tables, the groups must be isomorphic! He conjectured that the same holds true if the tables agree on 75% of their entries. I do not think the conjecture has been proved yet. See also Groups St. Andrews 2001 at Oxford, featuring the paper in Vol. 1, p. 143, of Drápal On the distance of 2-groups and 3-groups.

See also A. Drápal, How far apart can the group multiplication tables be?, European J. Combin. 13 (1992), no. 5, 335–343.

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    $\begingroup$ See my earlier MSE question: math.stackexchange.com/q/4810756 The upper and lower bounds are already improved to 83.3% and 77.8% respectively. $\endgroup$
    – Edward H
    Feb 18 at 6:02

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