on the set of distinct pairs of . See 2-transitive groups for more information about this action. Double cosets are important in connection with representation theory, when a representation of is used to construct an induced representation of , which is then restricted to . The corresponding double coset structure carries information about how the resulting representation decomposes. In the case of finite groups, this is Mackey's decomposition theorem. They are also important in functional analysis, where in some important cases functions left-invariant and right-invariant by a subgroup can form a commutative ring under convolution: see Gelfand pair. In geometry, a Clifford–Klein form is a double coset space , where is a reductive Lie group, is a closed subgroup, and is a discrete subgroup (of ) that acts properly discontinuously on the homogeneous space . In number theory, the Hecke algebra corresponding to a congruence subgroup of the modular group is spanned by elements of the double coset space ; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators corresponding to the double cosets or , where (these have different properties depending on whether and are coprime or not), and the diamond operators given by the double cosets where and we require (the choice of does not affect the answer). References Group theory