  Let F_q be the finite field with q elements. Consider the standard embedding
GL(n,F_q) -> GL(n+1,F_q). In this paper we prove that for every irreducible
representation pi of GL(n+1,F_q) over algebraically closed fields of
characteristic different from 2 we have dim\pi^GL(n,F_q)<=2.
  To do that we define a property of weak Gelfand pair and prove a
generalization of Gelfand trick for weak Gelfand pairs, using the
anti-involution transpose to get the result for GL(n+1,F_q),GL(n,F_q). In a
similar manner we show that for q not a power of 2 O(n+1,F_q),O(n,F_q) is a
Gelfand pair over algebraically closed fields of characteristic different from
2.
