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On Socratica, I saw a video demonstrating writing groups by writing the Cayley's table satisfying three conditions of the desired order. (1) Neutral element row and column are copies of the row and column headers. (2) Every row and column has neutral element once (3) All the elements of the set are present in each row and column.

My question is does this always lead to a group? I think the conditions are necessary but insufficient because they do not test the associativity.

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For example, this table is not a group table (where the rows and columns correspond to $e,a,b,c,d$): $$ \left[ \begin {array}{ccccc} e&a&b&c&d\\ a&e&c&d&b \\ b&c&d&a&e\\ c&d&e&b&a \\ d&b&a&e&c\end {array} \right] $$ since $(a a) b = eb = b$ while $a (ab) = ac = d$.

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have a look at the Associativity section in the Wikipedia page for Cayley table. It is not necessary for associativity to be a given in a Cayley table, as it can sometimes be used to characterize a quasigroup (fulfilling the other axioms for a group except for associativity).

The test for associativity requires three elements while the Cayley table only shows the results for two. Though you can probably generate a separate Cayley table for each of the results, at that point, you'll just be tackling the problem with brute force, and is impractical at best.

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