Let $G$ be a group and $n$ a positive integer. We say $G$ has Property $P(n)$ if, for every subset $F \subseteq G$ of size $n$, there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \ne id$ for all $x \in F \smallsetminus \{e\}$. Every group has $P(1)$ by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$; it is easy to see that some groups do not have $P(3)$. We provide an algebraic characterization of the countable groups (finite or infinite) that have $P(n)$. We deduce that if a countable group $G$ has $P(n-1)$ but does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. 查看全文>>