Inductive lattices of totally composition formations

Let τ be a subgroup functor such that all subgroups of every finite group G contained in τ(G) are subnormal in G. In this paper, we give a simple proof of the fact that the lattice of all τ -closed totally composition formations of finite groups is inductive.


Introduction
All groups considered in this paper are finite. A class of groups is a collection of groups satisfying the property that if a group G belongs to the collection, then every group isomorphic to G is also in the collection.
If a class of groups is a formation, it is closed with respect to forming quotient groups and subdirect products. This notion introduced by Gaschütz [3] in 1963 immediately became an object of extensive investigations. Saturated formations are very important in group theory; composition formations form a broader family of formations. By Baer's theorem, composition formations are precisely solvably saturated formations [2, p. 373].
Skiba [10] introduced the concept of an inductive lattice of formations in order to adapt lattice-theoretical methods for the investigation of saturated formations. This concept plays an important role in the research of the lattices of formations and their law systems (see Chapter 4 of the book [10], Chapter 4 of the book [19]; and the papers [5,6,7,8,12,13,14,18,20,21]).
Let Θ be a complete lattice of formations. A satellite f is called Θ-valued if all its values belong to Θ. We denote by Θ c the set of all formations having a composition Θ-valued satellite. In [11, p. 901], it is shown that this set is a complete lattice of formations.
A complete lattice Θ c is called inductive if for any collection of formations where f i is an integrated satellite of F i ∈ Θ c , the following equality holds: The inductance of a lattice Θ c , in fact, means that a research of the operation ∨ Θ c on the set Θ c can be reduced to a research of the operation ∨ Θ on the set Θ. Therefore, the inductance is one very useful property of the lattice Θ c .
Vorob'ev [17] proved that the lattice of all totally saturated formations is inductive. Moreover, it is already known that the lattice of all multiply composition formations is inductive (see [16]). However, the following question was still open. Question. Is the lattice of all totally composition formations inductive?
The aim of the present paper is to give a simple proof of the following theorem which gives a positive answer to this question. Theorem 1.1. The lattice of all τ -closed totally composition formations c τ ∞ is inductive.

Terminologies and notations
All unexplained notations and terminologies are standard. The reader is referred to [1,2,4,11] if necessary.

Subgroup functor τ
In various applications of the theory of classes of finite groups, it is often necessary to use formations closed with respect to some subgroup systems. Skiba [10] introduced the concept of a subgroup functor, which covers all the systems of subgroups under consideration.
In each group G, we select a system of subgroups τ (G). We say that τ is a subgroup functor if (1) G ∈ τ (G) for every group G; (2) for every epimorphism ϕ : A → B, and each H ∈ τ (A) and T ∈ τ (B), we have H ϕ ∈ τ (B) and T ϕ −1 ∈ τ (A).
If τ (G) = {G}, then the functor τ is called trivial. A formation F is called τ -closed if τ (G) ⊆ F for every group G of F (see [10]).
We consider only subgroup functors τ such that for every group G all subgroups of τ (G) are subnormal in G.

Composition formations
The set of all primes is denoted by P. Let p ∈ P, and G a group. Then the subgroup C p (G) is the intersection of the centralizers of all the abelian p-chief factors of G, with C p (G) = G if G has no abelian p-chief factors.
For every collection of groups X, we write Com(X) to denote the class of all groups L such that L is isomorphic to some abelian composition factor of some group in X. If X is the set of one group G, then we write Com(G) instead of Com(X).
The symbol R(G) denotes the product of all solvable normal subgroups of G. We consider a function f of the form and the class of groups If F is a formation such that F = CLF (f ) for a function f of the form ( * ), then F is said to be composition (solvably saturated) formation, and f is said to be a composition satellite of F (see [4, p. 4]).
If the values of composition satellites of some formation are themselves composition formations, then this circumstance leads to the following natural definition. Every formation is 0-multiply composition; for n > 0, a formation F is called n-multiply composition if F = CLF (f ), and all nonempty values of f are (n − 1)-multiply composition formations (see [11]).
A formation is called totally composition if it is n-multiply composition for all positive integers n.

Lattices of formations
A set of formations Θ is called a complete lattice of formations if the intersection of every set of formations in Θ belongs to Θ, and there is a formation F in Θ such that M ⊆ F for every other formation M of Θ (see [10]).
Every complete lattice of formations is a complete lattice in the ordinary sense. Various collections of formations form complete lattices; for example, the set of all saturated formations [10, p. 151], and the set of all composition (solvably saturated) formations [9, p. 97] are complete lattices of formations. Moreover for all positive integers n, the set of all n-multiply composition formations c n , and the set of all totally composition formations c ∞ = ∞ n=1 c n are complete lattices of formations (see [11, p. 904

]).
A formation in Θ is called a Θ-formation. Let Θ be a complete lattice of formations, and let {F i | i ∈ I} be an arbitrary collection of Θ-formations. We denote In particular, we write ∨ τ for all a ∈ P ∪ {0}.

Preliminaries
Following the paper [11], we set for every collection of groups X: We recall that the symbol N p denotes the class of all p-groups. Let Proof. The inclusion (c τ ∞ ) c ⊆ c τ ∞ is obvious. Let F ∈ c τ ∞ and F be a canonical composition satellite of F. Then by Lemmas 3.1 and 3.2 for all a ∈ P ∪ {0} and each positive integer n, the formation F (a) is τ -closed n-multiply composition. Thus, the satellite F is c τ ∞ -valued. Consequently, F ∈ (c τ ∞ ) c , and we have  Let {f i | i ∈ I} be a collection of composition satellites. Then by i∈I f i , we denote the composition satellite f such that f (a) = i∈I f i (a) for all a ∈ P∪{0} (see [11]).
Let {f i | i ∈ I} be the collection of all composition c τ ∞ -valued satellites of a formation F. Since the lattice c τ ∞ is complete using Lemma 3.6, we conclude that f = i∈I f i is a composition c τ ∞ -valued satellite of F. The satellite f is called minimal.
Let Θ be a complete lattice of formations. Then ΘformX is the intersection of all Θ-formations containing a collection of groups X. Thus, c τ ∞ formX is the intersection of all τ -closed totally composition formations containing a collection of groups X. The next lemma immediately follows from [11,Lemma 5] by Corollary 3.3, and gives a description of the minimal c τ ∞ -valued satellite of a formation c τ ∞ formX. Lemma 3.7. Let X be a nonempty collection of groups, F = c τ ∞ formX, π = π(Com(X)), and let f be the minimal c τ ∞ -valued composition satellite of F. Then the following statements hold: By Lemma 3.7, it is easy to show the following assertion. If F = CLF (f ) and f (a) ⊆ F for all a ∈ P ∪ {0}, then f is called an integrated satellite of F.

Inductance of the lattice c τ ω∞
Proof of Theorem. Let {F i | i ∈ I} be a collection of τ -closed totally composition formations, and f i be an integrated c τ ∞ -valued composition satellite of F i . Let .
We shall show that F = M proceeding by induction on i.
Step 1. Let i = 2, p ∈ P, and h j be the minimal c τ ∞ -valued composition satellite of the formation F j = CLF (f j ), where j = 1, 2. Then by Corollary 3.5, we have where F is the canonical c τ ∞ -valued composition satellite of the formation F. Then by Lemma 3.7, we have Step 2. Let i > 2, and the assertion is true for i = r − 1 by induction. Then This proves the theorem.
Each complete sublattice of the inductive lattice is an inductive lattice. Thus, we have the following result. If τ is trivial, we have the following result.    (f 1 , ..., f m )).

Some applications
Let A be a group, and p be a prime. We use Z p A to denote the regular wreath product of groups Z p and A (see [2, p. 66]). The following lemma is proved by direct calculation. We shall show that h(p) = f (p) by induction on the number r of occurrences of the symbols in {∩, ∨ τ ∞ } into ξ. The case r = 1 holds using Lemmas 5.1 and 5.2.