# Can we define a group from its Cayley's table?

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.

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 $a (ab) = ac = d$.