Consider the set $A = \{a_1,a_2,...,a_n,e\}$. Here, $e$ will be our identity. Define the operation * such that $a_i*a_j = e$ for all $i,j\in\{1,2,...,n\}$ and $k*e = e*k = k$ for all $k\in A$. Is this a group?
I feel that it satisfies all three properties: associativity, existence of an identity, existence of an inverse for each element. However, clearly the inverses here are not unique... which goes against the easily derivable fact that inverses are unique in groups. What's going on intuitively here?
EDIT: Oh wow, it's not associative, thanks guys.