In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory. When G is a finite group the simplest definition is, roughly speaking, that the (K,K)-double cosets in G commute. More precisely, the Hecke algebra, the algebra of functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G. In general, the definition of Gelfand pair is roughly that the restriction to K of any irreducible representation of G contains the trivial representation of K with multiplicity no more than 1. In each case one should specify the class of considered representations and the meaning of contains. Definitions In each area, the class of representations and the definition of containment for representations is slightly different. Explicit definitions in several such cases are given here. Finite group case When G is a finite group the following are equivalent (G,K) is a Gelfand pair. The algebra of (K,K)-double invariant functions on G with multiplication defined by convolution is commutative. For any irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional. For any irreducible representation π of G, the dimension of HomK(π, C) is less than or equal to 1, where C denotes the trivial representation. The permutation representation of G on the cosets of K is multiplicity-free, that is, it decomposes into a direct sum of distinct absolutely irreducible representations in characteristic zero. The centralizer algebra (Schur algebra) of the permutation representation is commutative. (G/N, K/N) is a Gelfand pair, where N is a normal subgroup of G contained in K. Compact group case When G is a compact topological group the following are equivalent: (G,K) is a Gelfand pair. The algebra of (K,K)-double invariant compactly supported continuous measures on G with multiplication defined by convolution is commutative. For any continuous, locally convex, irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional. For any continuous, locally convex,