Normal structure of finite groups with arithmetic restrictions on their subgroups .

A finite group H is called a P-group for some set of prime numbers P when all primes that divide the order |H| lie in P. The P-subgroup H in G is its Hall subgroup iff besides this the index |G:H| is not divided by any prime p from the set P. We will focus on the structure of normal series in finite groups with Hall maximal subgroups and in finite groups which are prime spectrum minimal. Some additional properties of such groups will also be discussed.