On one generalization of Baer's theorems about hypercenter and nilpotent residual

Let $\mathfrak{F}$ be a class of finite groups which are the direct products of their Hall $\pi_i$-subgroups corresponding to a given partition $\sigma=\{\pi_i|i\in I, i\ne j\rightarrow\pi_i\cap\pi_j=\varnothing\}$ of a nonempty subset $\pi$ of the set of all primes. This class is a local formation. In this paper the properties of $\mathfrak{F}$-hypercenter and $\mathfrak{F}$-residual of a finite group are studied. It was shown that for a finite $\pi$-group $G$ the intersection of all normalizers of all maximal $\pi_i$-subgroups for all $i$ is the $\mathfrak{F}$-hypercenter of $G$. As corollaries were obtained well-known properties of hypercenter and nilpotent residual of finite groups.