In the group theory, ”arithmetical” properties of a group G are the properties which are defined by some arithmetical parameters of G.

Let G be a finite group. The spectrum of G is the set omega(G) of orders of all its elements. The subset pi(G) of prime elements of omega(G) is called prime spectrum of G. The spectrum omega(G) of a group G defines its prime graph (or Grünberg–Kegel graph) Gamma(G) with vertex set pi(G), in which any two different vertices p and q are adjacent if and only if the number pq belongs to the set omega(G). The spectrum, the prime spectrum and the prime graph of a finite group G give examples of arithmetical parameters of G.

Such invariants of a group G as sets of composition and chief factors of G with details of action of G on its chief factors are called the ”normal structure” of G . Cross impact of arithmetical properties of a finite group and its normal structure is well known.

In the present talk we will discuss some questions on the finite prime spectrum minimal, prime graph minimal and spectrum minimal groups. In particular, we will describe all the cases when the prime graphs of a finite simple group and of its proper subgroup coincide and spectrums of a finite simple group and of its proper subgroup coincide.