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Throughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, and $σ(\mathfrak{F})=\cup _G{}_\in{}_\mathfrak{F}σ(G)$. A function $f$ of the form  $f\colon σ\to$ {formations of groups} is called a formation σ-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows: $LF_{\sigma}(f)=(G$ is a group $|G=1$ или $G\ne1$ and $G/O_σ{}'_i{}_,{}_σ{}_i{}(G)\in f(σ_i{})$ for all $σ_i{}\in σ(G))$. If for some formation $σ$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f)$, then the class $\mathfrak{F}$ is called $σ$-local definition of $\mathfrak{F}$. Every formation is called 0-multiply $σ$-local. For $n$ > 0, a formation $\mathfrak{F}$ is called $n$-multiply $σ$-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f)$, where $f(σ_i{})$ is $(n – 1)$-multiply $σ$-local for all $σ_i{}\in σ(\mathfrak{F})$. A formation is called totally $σ$-local if it is $n$-multiply $σ$-local for all non-negative integer $n$. The aim of this paper is to study properties of the lattice of totally $σ$-local formations. In particular, we prove that the lattice of all totally $σ$-local formations is algebraic and distributive.

Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation. We denote by \mathbf{Z}_{\mathfrak{F}}(G) the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor H/K of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is, (H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F}. A subgroup A\leqslant G is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G such that either A_{i-1}\trianglelefteq A_{i} or A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F} for all i=1,\ldots,n. In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let 𝔉 be a hereditary saturated formation containing all nilpotent groups, and let 𝑆 be a 𝐾-𝔉-subnormal subgroup of 𝐺. If \mathbf{Z}_{\mathfrak{F}}(E)=1 for every subgroup 𝐸 of 𝐺 such that S\leqslant E, then \mathbf{C}_{G}(D)\leqslant D, where D=S^{\mathfrak{F}} is the 𝔉-residual of 𝑆.

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