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For a group, arbitrary its subgroup and a transversal to this subgroup, using the binary operation of the group, by a standard construction four induced mappings are determined, some their properties are established and a concept of hypergroups over the group is introduced, which generalizes the concept of quotient-groups. By a construction, called the exact product, associated with a hypergroup over the group (which generalizes the semidirect product of two groups), a universal property of the standard construction is proved. It is proved that the hypergroups over the group generalize also the fields and the linear spaces over the field. A necessary and sufficient condition for a group to be the multiplicative group of a field is established in the language of hypergroups over the group. 

transversal, induced structure, hypergroup over the group.