Finite Groups and Families of Modular Forms Associated with Them

In the paper, a correspondence between finite groups and $q$-series which, under specialization, are Fourier expansions of modular forms is studied. The categories thus arising are investigated. The problem of describing groups to which $q$-series with multiplicative coefficients correspond is considered. Subgroups of this kind are contained in any group. The notion of modular analog of the genetic code of a group is introduced and studied.