Now the Gelfand property for the pair (G,K) is an analog of the Schur's lemma. Using this approach one can take any concepts of representation theory and generalize them to the case of spherical pair. For example, the relative trace formula is obtained from the trace formula by this procedure. Examples Finite groups A few common examples of Gelfand pairs are: (Sym(n+1), Sym(n)), the symmetric group acting on n+1 points and a point stabilizer that is naturally isomorphic to on n points. (AGL(n, q), GL(n, q)), the affine (general linear) group and a point stabilizer that is naturally isomorphic to the general linear group. If (G,K) is a Gelfand pair, then (G/N,K/N) is a Gelfand pair for every G-normal subgroup N of K. For many purposes it suffices to consider K without any such non-identity normal subgroups. The action of G on the cosets of K is thus faithful, so one is then looking at permutation groups G with point stabilizers K. To be a Gelfand pair is equivalent to for every χ in Irr(G). Since by Frobenius reciprocity and is the character of the permutation action, a permutation group defines a Gelfand pair if and only if the permutation character is a so-called multiplicity-free permutation character. Such multiplicity-free permutation characters were determined for the sporadic groups in . This gives rise to a class of examples of finite groups with Gelfand pairs: the 2-transitive groups. A permutation group G is 2-transitive if the stabilizer K of a point acts transitively on the remaining points. In particular, G the symmetric group on n+1 points and K the symmetric group on n points forms a Gelfand pair for every n≥1. This follows because the character of a 2-transitive permutation action is of the form 1+χ for some irreducible character χ and the trivial character 1, . Indeed, if G is a transitive permutation group whose point stabilizer K has at most four orbits (including the trivial orbit containing only the stabilized point), then its Schur ring is commutative and (G,K) is a Gelfand pair, . If G is a primitive group of degree twice a prime with point stabilizer K, then again (G,K) is a Gelfand pair, . The Gelfand pairs (Sym(n),K) were classified in . Roughly speaking, K must be contained as a subgroup of small index in one of the following groups unless n is smaller than 18: Sym(n