On Relations Associated with the Euler Function

The paper studies the properties of the set of numbers smaller than and coprime to \( n \) with the modulo \( n \) multiplication operation introduced on it (this object is sometimes called the Euler group). The cardinality of such a set is the well-known Euler function \( \varphi (n) \), which is one of the classical functions in the number theory. The fields of its application are quite wide and include, for example, various branches of discrete mathematics, and it also has significant applications in cryptography. The paper considers various combinatorial problems arising in the study of the Euler group and the Euler function. Relations between theoretical and numerical parameters associated with the Euler group and Euler function are derived. The combinatorial relations obtained in the paper can be used when solving applied combinatorial problems and in cryptography.