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 formula_8 for every "χ" in Irr("G"). Since formula_9 by Frobenius reciprocity and formula_10 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 .