2505/2505.00262.md

Title: Certain residual properties of HNN-extensions with normal associated subgroups

URL Source: https://arxiv.org/html/2505.00262

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Why HTML? Report Issue Back to Abstract Download PDF Abstract 1Introduction 2Statement of results 3Some auxiliary concepts and facts 4Proof of Theorems 1–2 and Corollary 1 5Proof of Theorem 3 6Proof of Theorems 4–6 and Corollary 2 References License: arXiv.org perpetual non-exclusive license arXiv:2505.00262v2 [math.GR] 02 May 2025 Certain residual properties of HNN-extensions with normal associated subgroups E. V. Sokolov and E. A. Tumanova Ivanovo State University, Russia ev-sokolov@yandex.ru, helenfog@bk.ru Abstract.

Let 𝔼 be the HNN-extension of a group  𝐵 with subgroups  𝐻 and  𝐾 associated according to an isomorphism 𝜑 : 𝐻 → 𝐾 . Suppose that 𝐻 and  𝐾 are normal in  𝐵 and  ( 𝐻 ∩ 𝐾 ) ⁢ 𝜑

𝐻 ∩ 𝐾 . Under these assumptions, we prove necessary and sufficient conditions for  𝔼 to be residually a  𝒞 -group, where 𝒞 is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual 𝑝 -finiteness of the group  𝔼 .

Key words and phrases: Residual properties, residual finiteness, residual 𝑝 -finiteness, residual solvability, root class of groups, HNN-extension The study was supported by the Russian Science Foundation grant No. 24-21-00307, http://rscf.ru/en/project/24-21-00307/ 1.Introduction

Let 𝒞 be a class of groups. Following [1], we say that a group  𝑋 is residually a  𝒞 -group if any of its non-trivial elements is mapped to a non-trivial element by a suitable homomorphism of  𝑋 onto a  𝒞 -group (i.e., a group from the class  𝒞 ). Recall that, if  𝒞 is the class of all finite groups (or all solvable groups, or finite 𝑝 -groups, where 𝑝 is a prime), then a residually 𝒞 -group is also referred to as a residually finite (respectively, residually solvable, residually 𝑝 -finite) group. We therefore use the term ‘‘residual 𝒞 -ness’’ along with the well-known notions of residual finiteness, residual 𝑝 -finiteness, and residual solvability. Let us clarify that the residual 𝒞 -ness of a group  𝑋 is the same as the property of  𝑋 to be residually a  𝒞 -group. In Section 1, for brevity, this term will also assume that 𝒞 is a root class of groups.

According to [2, 3], a class of groups  𝒞 is called a root class if it contains non-trivial groups, is closed under taking subgroups, and satisfies any of the following conditions, the equivalence of which is proved in [4]:

1)  for every group  𝑋 and for every subnormal series 1 ⩽ 𝑍 ⩽ 𝑌 ⩽ 𝑋 whose factors 𝑋 / 𝑌 and  𝑌 / 𝑍 belong to  𝒞 , there exists a normal subgroup  𝑇 of  𝑋 such that 𝑋 / 𝑇 ∈ 𝒞 and  𝑇 ⩽ 𝑍 (Gruenberg’s condition);

2)  the class  𝒞 is closed under taking unrestricted wreath products;

3)  the class  𝒞 is closed under taking extensions and, together with any two groups  𝑋 and  𝑌 , contains the unrestricted direct product ∏ 𝑦 ∈ 𝑌 𝑋 𝑦 , where 𝑋 𝑦 is an isomorphic copy of  𝑋 for each 𝑦 ∈ 𝑌 .

Examples of root classes are the classes of all finite groups, finite 𝑝 -groups (where 𝑝 is a prime), periodic 𝔓 -groups of finite exponent (where 𝔓 is a non-empty set of primes), all solvable groups, and all torsion-free groups. It is also easy to see that the intersection of a family of root classes is again a root class if it contains a non-trivial group. The use of the concept of a root class turned out to be very productive in studying the residual properties of free constructions of groups: free and tree products, HNN-extensions, fundamental groups of graphs of group, etc. Clearly, it enables us to prove several statements at once instead of just one. But what is more important, the facts on residual 𝒞 -ness, where 𝒞 is an arbitrary root class of groups, and the methods for finding them are well compatible with each other. This allows one to easily move from one free construction to another and quickly complicate the groups under consideration (see, e.g., [5, 6, 7, 8, 9, 10, 11]).

When studying the residual 𝒞 -ness of a group-theoretic construction, the main question is whether the construction inherits this property from the groups that compose it. As a rule, this question can only be answered by imposing various restrictions on the above-mentioned groups and their subgroups. The goal of this paper is to find necessary and sufficient conditions for the residual 𝒞 -ness of an HNN-extension whose associated subgroups are normal in the base group (here and below we follow [12] in the use of terms related to HNN-extensions).

Considering the known results on the residual 𝒞 -ness of HNN-extensions, one can observe that most of them concern the case of residual finiteness [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The papers [26, 27, 28, 29, 30, 31] give some facts on the residual 𝑝 -finiteness of this construction. In [32, 33, 34, 5, 6, 7, 35, 36, 37, 38, 11], the residual 𝒞 -ness of HNN-extensions is studied provided 𝒞 is an arbitrary root class of groups, which possibly satisfies some additional restrictions. It should be noted that, at the time of writing these lines, the results of the last listed papers generalize all known facts on the residual solvability and residual 𝔓 -finiteness of HNN-extensions, where 𝔓 is a non-empty set of primes, as above.

The main method for studying the residual 𝒞 -ness of free constructions of groups is the so-called ‘‘filtration approach’’. It was originally proposed in [39] to study the residual finiteness of the free product of two groups with an amalgamated subgroup. After a number of generalizations and adaptations [14, 40, 41, 42, 28, 43], this approach was extended in [44] to the case of an arbitrary root class  𝒞 and the fundamental group of an arbitrary graph of groups. The method includes two steps and, when applied to HNN-extensions, can be described as follows.

The first step is to find conditions for an HNN-extension  𝔼 to have a homomorphism onto a  𝒞 -group that acts injectively on the base group. The existence of such a homomorphism is sometimes equivalent to the residual 𝒞 -ness of  𝔼 . For example, this equivalence holds if 𝒞 consists of finite groups. But in general this is not the case [45, 46].

We call the assertions proved at this step the first-level results. It should be noted that there are no general approaches to finding them and each new fact of this type is very valuable. Examples of such assertions are the theorem on the residual finiteness of an HNN-extension of a finite group [14] and the criteria for the residual 𝑝 -finiteness of the same construction [26, 27, 28, 31]. When 𝒞 is an arbitrary root class of groups closed under taking quotient groups, first-level results are found for

–

an HNN-extension with coinciding associated subgroups that are normal in the base group [33];

–

a number of HNN-extensions in which at least one of the associated subgroups lies in the center of the base group [7, 37, 11].

In this paper, we supplement this list and prove a criterion for the existence of a homomorphism with the properties described above provided the associated subgroups of an HNN-extension  𝔼 are normal in the base group, while their intersection is normal in  𝔼 (see Theorem 1 below).

Omitting technical details, we can say that the second step of the method consists in finding a sufficiently large number of homomorphisms mapping the HNN-extension  𝔼 onto HNN-extensions satisfying the conditions of the previously gotten first-level results. This allows us to prove the residual 𝒞 -ness of  𝔼 when the base group does not necessarily belong to the class  𝒞 . We call the sufficient conditions of the residual 𝒞 -ness found at this step the second-level results. As a rule, they can be proved only under restrictions stronger than those at step 1. This is illustrated, in particular, by many years of studying the property of residual finiteness, during which a universal criterion for the residual finiteness of an HNN-extension of an arbitrary residually finite group was never found. In the present paper, the second-level results, Theorems 4–6, are also proved under certain assumptions supplementing the conditions of the criterion given by Theorem 1.

2.Statement of results

In what follows, the expression 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ means that 𝔼 is the HNN-extension of a group  𝐵 with a stable letter  𝑡 and subgroups  𝐻 and  𝐾 associated according to an isomorphism 𝜑 : 𝐻 → 𝐾 . Let us say that the group  𝔼 satisfies  ( ∗ )  if

1)  the subgroups  𝐻 and  𝐾 are normal in  𝐵 ;

2)  the subgroup 𝐿

𝐻 ∩ 𝐾 is  𝜑 -invariant, i.e.,  𝜑 | 𝐿 ∈ Aut ⁡ 𝐿 .

If 𝑋 is a group and  𝑌 is a normal subgroup of  𝑋 , then the restriction to  𝑌 of any inner automorphism of  𝑋 is an automorphism of  𝑌 . The set of all such automorphisms is a subgroup of  Aut ⁡ 𝑌 , which we denote below by  Aut 𝑋 ⁡ ( 𝑌 ) . Let us note that, if  𝔼 satisfies  ( ∗ ) , then the following is defined:

a)

the subgroups ℌ

Aut 𝐵 ⁡ ( 𝐻 ) , 𝔎

𝜑 ⁢ Aut 𝐵 ⁡ ( 𝐾 ) ⁢ 𝜑 − 1 , and  𝔘

sgp ⁡ { ℌ , 𝔎 } of  Aut ⁡ 𝐻 ;

b)

the subgroups 𝔏

Aut 𝐵 ⁡ ( 𝐿 ) , 𝔉

sgp ⁡ { 𝜑 | 𝐿 } , and  𝔙

sgp ⁡ { 𝔏 , 𝔉 } of  Aut ⁡ 𝐿 .

Throughout the paper, it is assumed that, if 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ , then the symbols  𝐿 ,  ℌ , 𝔎 , 𝔘 , 𝔏 , 𝔉 , and  𝔙 are defined as above.

Theorem 1.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞 is a root class of groups closed under taking quotient groups. If  𝐵 ∈ 𝒞 , then the following statements are equivalent and any of them implies that 𝔼 is residually a  𝒞 -group.

1.  There exists a homomorphism of  𝔼 onto a group from  𝒞 acting injectively on the subgroup  𝐵 .

2.  The inclusions 𝔘 , 𝔙 ∈ 𝒞 hold.

Theorem 1 admits the following generalization.

Theorem 2.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞 is a root class of groups closed under taking quotient groups. If  𝐵 has a homomorphism  𝜎 onto a group from  𝒞 acting injectively on the subgroup  𝐻 ⁢ 𝐾 , then the following statements hold.

1.  The condition 𝔘 , 𝔙 ∈ 𝒞 is equivalent to the existence of a homomorphism of  𝔼 onto a group from  𝒞 that extends  𝜎 .

2.  If 𝔘 , 𝔙 ∈ 𝒞 and  𝐵 is residually a  𝒞 -group, then 𝔼 is also residually a  𝒞 -group.

Let us note that the assertions like Theorem 2 can be useful for proving new first-level results (see, e.g., [11]).

Corollary 1.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞 is a root class of groups closed under taking quotient groups, and the subgroups  𝐻 and  𝐾 are finite. Then 𝔼 is residually a  𝒞 -group if and only if 𝐵 is residually a  𝒞 -group and  𝔘 , 𝔙 ∈ 𝒞 .

Theorem 1 requires that 𝐵 belongs to  𝒞 . Theorem 2 removes this restriction, but still implicitly assumes that 𝐻 , 𝐾 ∈ 𝒞 . Now let us consider the case where 𝐵 , 𝐻 , and  𝐾 are arbitrary residually 𝒞 -groups. We start with some necessary conditions for  𝔼 to be residually a  𝒞 -group.

Theorem 3.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a class of groups closed under taking subgroups, quotient groups, and direct products of a finite number of factors. Suppose also that at least one of the following statements holds:

( 𝑎 )

𝔘

ℌ or  𝔘

𝔎 ;

( 𝑏 )

𝐻 and  𝐾 satisfy a non-trivial identity;

( 𝑐 )

ℌ satisfies a non-trivial identity;

( 𝑑 )

𝔎 satisfies a non-trivial identity.

If 𝔼 is residually a  𝒞 -group, then the quotient groups  𝐵 / 𝐻 and  𝐵 / 𝐾 have the same property.

Let us note that the papers [47, 44] contain several more necessary conditions for the residual 𝒞 -ness of HNN-extensions, which are similar to Theorem 3.

Given a class of groups  𝒞 and a group  𝑋 , we denote by  𝒞 ∗ ⁢ ( 𝑋 ) the family of normal subgroups of  𝑋 defined as follows: 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) if and only if 𝑋 / 𝑁 ∈ 𝒞 . Let us say that 𝑋 is  𝒞 -quasi-regular with respect to its subgroup  𝑌 if, for each subgroup 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝑌 ) , there exists a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) such that 𝑁 ∩ 𝑌 ⩽ 𝑀 . The property of  𝒞 -quasi-regularity is closely related to the classical notion of a  𝒞 -separable subgroup [48] and plays an important role in constructing the kernels of the homomorphisms that map a free construction of groups onto the groups from  𝒞 . Therefore, it is often a part of conditions sufficient for such a construction to be residually a  𝒞 -group (see, e.g., [17, 49, 11]). In [38, 50, 51], a number of situations are described in which a group turns out to be  𝒞 -quasi-regular with respect to its subgroup.

Theorem 4.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞 is a root class of groups closed under taking quotient groups, and at least one of the following statements holds:

( 𝛼 )

𝐻 / 𝐿 ∈ 𝒞 ;

( 𝛽 )

there exists a homomorphism of  𝐵 onto a group from  𝒞 acting injectively on  𝐿 .

Suppose also that 𝔘 , 𝔙 ∈ 𝒞 and  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 . If  𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups, then 𝔼 and  𝐵 are residually 𝒞 -groups simultaneously.

Let us note that, if  𝒞 is a root class of groups closed under taking quotient groups, while the HNN-extension  𝔼 is residually a  𝒞 -group and satisfies  ( ∗ ) , then, by Proposition 3.12 below, 𝑈 and  𝑉 are residually 𝒞 -groups and, therefore, ℌ , 𝔎 , 𝔏 , 𝔉 , 𝔘 , and  𝔙 belong to  𝒞 when they are finite. However, in general, neither the inclusions 𝔘 ∈ 𝒞 and  𝔙 ∈ 𝒞 , which appear in Theorems 1, 2, and 4, nor the weaker conditions ℌ ∈ 𝒞 , 𝔎 ∈ 𝒞 , 𝔏 ∈ 𝒞 , and  𝔉 ∈ 𝒞 are necessary for  𝔼 to be residually a  𝒞 -group, as the following example shows.

Example.

Suppose that

𝐸

⟨ 𝑎 , 𝑏 , 𝑐 , 𝑡 ; [ 𝑎 , 𝑏 ]

[ 𝑏 , 𝑐 ]

[ 𝑏 , 𝑡 ]

1 , 𝑐 − 1 𝑎 𝑐

𝑡 − 1 𝑎 𝑡

𝑎 𝑏 ⟩ ,

𝐴

sgp ⁡ { 𝑎 , 𝑏 } , and  𝜑 is the automorphism of  𝐴 taking  𝑎 to  𝑎 ⁢ 𝑏 and  𝑏 to  𝑏 . Then 𝐸 is the HNN-extension of the group

𝐵

⟨ 𝑎 , 𝑏 , 𝑐 ; [ 𝑎 , 𝑏 ]

[ 𝑏 , 𝑐 ]

1 , 𝑐 − 1 𝑎 𝑐

𝑎 𝑏 ⟩

with the coinciding subgroups 𝐻

𝐴

𝐾 associated according to  𝜑 . The group  𝐵 , in turn, is the extension of the free abelian group  𝐴 by the infinite cyclic group with the generator  𝑐 , the conjugation by which acts on  𝐴 as  𝜑 . Therefore,

Aut 𝐵 ⁡ ( 𝐻 )

Aut 𝐵 ⁡ ( 𝐾 )

Aut 𝐵 ⁡ ( 𝐻 ∩ 𝐾 )

sgp ⁡ { 𝜑 | 𝐻 ∩ 𝐾 }

is the infinite cyclic group generated by  𝜑 . At the same time, 𝐸 splits as the generalized free product of the isomorphic polycyclic groups  𝐵  and

𝐷

⟨ 𝑎 , 𝑏 , 𝑡 ; [ 𝑎 , 𝑏 ]

[ 𝑏 , 𝑡 ]

1 , 𝑡 − 1 𝑎 𝑡

𝑎 𝑏 ⟩

with the normal amalgamated subgroup  𝐴 . Hence, it is residually finite by Theorem 9 from [39].

The next two theorems (and Propositions 6.5–6.8 in Section 6) describe some cases where the condition 𝔘 , 𝔙 ∈ 𝒞 from Theorem 4 can be modified or weakened.

Theorem 5.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞 is a root class of groups closed under taking quotient groups, and at least one of Statements  ( 𝛼 ) and  ( 𝛽 ) from Theorem 4 holds. Suppose also that 𝔘

ℌ or  𝔘

𝔎 , 𝔙 ∈ 𝒞 , and  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 . Then 𝔼 is residually a  𝒞 -group if and only if the groups  𝐵 , 𝐵 / 𝐻 , and  𝐵 / 𝐾 have the same property.

Theorem 6.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞 is a root class of groups consisting only of periodic groups and closed under taking quotient groups. If  𝐻 and  𝐾 are locally cyclic groups, then the following statements hold.

1.  Let 𝐵 / 𝐻 and  𝐵 / 𝐾 be residually 𝒞 -groups. Then the group  𝐻 / 𝐿 is finite if and only if ( 𝛼 ) holds.

2.  Let 𝐵 be residually a  𝒞 -group. Then the group  𝐿 is finite if and only if ( 𝛽 ) holds.

3.  Suppose that 𝐻 / 𝐿 is finite, 𝔉 ∈ 𝒞 , and  𝐵 is  𝒞 -quasi-regular with respect to  𝐿 . Then 𝔼 is residually a  𝒞 -group if and only if the groups  𝐵 , 𝐵 / 𝐻 , and  𝐵 / 𝐾 have the same property.

4.  Suppose that 𝐿 is finite and  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 . Then 𝔼 is residually a  𝒞 -group if and only if 𝐵 , 𝐵 / 𝐻 , and  𝐵 / 𝐾 are residually 𝒞 -groups and  𝔉 ∈ 𝒞 .

Let us note that the condition ‘‘ 𝔘

ℌ or  𝔘

𝔎 ’’ holds if at least one of the subgroups  𝐻 and  𝐾 lies in the center of  𝐵 . In this case, the subgroup  𝐿 is certainly central in  𝐵 , whence 𝔏

1 and  𝔙

𝔉 . Therefore, Theorem 5 generalizes Theorem 5 from [11]. As a comment to Theorem 6, we also note that, if  𝑝 is a prime and  ℱ 𝑝 is the class of finite 𝑝 -groups, then every residually ℱ 𝑝 -group is  ℱ 𝑝 -quasi-regular with respect to any of its locally cyclic subgroups [51, Theorem 3].

Given a class of groups 𝒞 consisting only of periodic groups, let us denote by  𝔓 ⁢ ( 𝒞 ) the set of primes defined as follows: 𝑝 ∈ 𝔓 ⁢ ( 𝒞 ) if and only if there exists a  𝒞 -group  𝑍 such that 𝑝 divides the order of some element of  𝑍 . A subgroup  𝑌 of a group  𝑋 is said to be  𝔓 ⁢ ( 𝒞 ) ′ -isolated in this group if, for any element 𝑥 ∈ 𝑋 and for any prime 𝑞 ∉ 𝔓 ⁢ ( 𝒞 ) , it follows from the inclusion 𝑥 𝑞 ∈ 𝑌 that 𝑥 ∈ 𝑌 . Clearly, if  𝔓 ⁢ ( 𝒞 ) contains all prime numbers, then every subgroup is  𝔓 ⁢ ( 𝒞 ) ′ -isolated.

Following [38], we say that

–  an abelian group is  𝒞 -bounded if, for any quotient group  𝐵 of  𝐴 and for any 𝑝 ∈ 𝔓 ⁢ ( 𝒞 ) , the  𝑝 -power torsion subgroup of  𝐵 has a finite exponent and a cardinality not exceeding the cardinality of some 𝒞 -group;

–  a nilpotent group is  𝒞 -bounded if it has a finite central series with  𝒞 -bounded abelian factors.

It is easy to see that, if  𝒞 is a root class of groups consisting only of periodic groups, then every finitely generated abelian group is  𝒞 -bounded abelian and, therefore, all finitely generated nilpotent groups are  𝒞 -bounded nilpotent.

Corollary 2.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups consisting only of periodic groups and closed under taking quotient groups. Suppose also that 𝐵 is a  𝒞 -bounded nilpotent group and at least one of Statements  ( 𝛼 ) and  ( 𝛽 ) holds. Finally, let at least one of the following statements hold:

( 𝑎 )

𝔘 , 𝔙 ∈ 𝒞 ;

( 𝑏 )

𝔘

ℌ or  𝔘

𝔎 , and  𝔙 ∈ 𝒞 ;

( 𝑐 )

𝐻 and  𝐾 are locally cyclic subgroups and  𝔉 ∈ 𝒞 .

Then 𝔼 is residually a  𝒞 -group if and only if the subgroups  { 1 } , 𝐻 , and  𝐾 are  𝔓 ⁢ ( 𝒞 ) ′ -isolated in  𝐵 .

Let us note that the known results on the residual finiteness and residual 𝑝 -finiteness of HNN-extensions do not generalize the assertions that follow from Theorems 4–6 and Corollary 2 when 𝒞 is the class of all finite groups or finite 𝑝 -groups. Thus, these assertions are also of interest. Proofs of the formulated theorems and corollaries are given in Sections 4–6.

3.Some auxiliary concepts and facts

We use the following notations throughout the paper:

⟨ 𝑥 ⟩

the cyclic group generated by an element  𝑥 ;

𝑥 ^

the inner automorphism produced by an element  𝑥 ;

[ 𝑥 , 𝑦 ]

the commutator of elements  𝑥 and  𝑦 , which is equal to  𝑥 − 1 ⁢ 𝑦 − 1 ⁢ 𝑥 ⁢ 𝑦 ;

[ 𝑋 : 𝑌 ]

the index of a subgroup  𝑌 in a group  𝑋 ;

ker ⁡ 𝜎

the kernel of a homomorphism  𝜎 ;

Im ⁡ 𝜎

the image of a homomorphism  𝜎 .

Let 𝒞 be a class of groups, and let 𝑋 be a group. Following [48], we say that a subgroup  𝑌 of a group  𝑋 is  𝒞 -separable in this group if, for each element 𝑥 ∈ 𝑋 ∖ 𝑌 , there exists a homomorphism  𝜎 of  𝑋 onto a group from  𝒞 such that 𝑥 ⁢ 𝜎 ∉ 𝑌 ⁢ 𝜎 .

Proposition 3.1.

[52, Proposition 3] Suppose that 𝒞 is a class of groups closed under taking quotient groups, 𝑋  is a group, and  𝑌 is a normal subgroup of  𝑋 . Then 𝑌 is  𝒞 -separable in  𝑋 if and only if 𝑋 / 𝑌 is residually a  𝒞 -group.

Proposition 3.2.

[52, Proposition 4] Suppose that 𝒞 is a class of groups closed under taking subgroups, 𝑋  is a group, 𝑌  is a subgroup of  𝑋 , and  𝑍 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) . Then 𝑌 ∩ 𝑍 ∈ 𝒞 ∗ ⁢ ( 𝑌 ) and, if  𝑋 is residually a  𝒞 -group, then 𝑌 is also residually a  𝒞 -group.

Proposition 3.3.

[52, Proposition 2] Suppose that 𝒞 is a class of groups closed undertaking subgroups and direct products of a finite number of factors. Then, for every group  𝑋 , the following statements hold.

1.  The intersection of finitely many subgroups of the family  𝒞 ∗ ⁢ ( 𝑋 ) is again a subgroup of this family.

2.  If 𝑋 is residually a  𝒞 -group and  𝑌 is a finite subgroup of  𝑋 , then there exists a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) that meets 𝑌 trivially, whence 𝑌 ∈ 𝒞 .

Proposition 3.4.

[53, Proposition 4] Suppose that 𝒞 is a class of groups closed under taking quotient groups, 𝑋  is a group, and  𝑌 is a normal subgroup of  𝑋 . If there exists a homomorphism of  𝑋 onto a group from  𝒞 acting injectively on  𝑌 , then Aut 𝑋 ⁡ ( 𝑌 ) ∈ 𝒞 .

Proposition 3.5.

If 𝒞 is a class of groups closed under taking quotient groups, 𝑋  is residually a  𝒞 -group, and  𝑌 is a normal subgroup of  𝑋 , then Aut 𝑋 ⁡ ( 𝑌 ) is also residually a  𝒞 -group.

Proof.

It is easy to see that Aut 𝑋 ⁡ ( 𝑌 ) ≅ 𝑋 / 𝒵 𝑋 ⁢ ( 𝑌 ) , where 𝒵 𝑋 ⁢ ( 𝑌 ) is the centralizer of  𝑌 in  𝑋 . Due to Proposition 3.1, it suffices to show that, if  𝒵 𝑋 ⁢ ( 𝑌 ) ≠ 𝑋 , then the subgroup  𝒵 𝑋 ⁢ ( 𝑌 ) is  𝒞 -separable in  𝑋 .

Let 𝑥 ∈ 𝑋 ∖ 𝒵 𝑋 ⁢ ( 𝑌 ) . Then [ 𝑥 , 𝑦 ] ≠ 1 for some 𝑦 ∈ 𝑌 . Since 𝑋 is residually a  𝒞 -group, there exists a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) which does not contain the commutator  [ 𝑥 , 𝑦 ] . It follows that 𝑥 ∉ 𝒵 𝑋 ⁢ ( 𝑌 ) ⁢ 𝑁 and, hence, the subgroup  𝒵 𝑋 ⁢ ( 𝑌 ) is  𝒞 -separable. ∎

The proof of the following proposition is quite simple and is therefore omitted.

Proposition 3.6.

Suppose that 𝑋 is a group, 𝑌  is a normal subgroup of  𝑋 , and  𝜎 is a homomorphism of  𝑋 . Suppose also that 𝜎 ¯ : Aut 𝑋 ⁡ ( 𝑌 ) → Aut 𝑋 ⁢ 𝜎 ⁡ ( 𝑌 ⁢ 𝜎 ) is the map taking  𝑥 ^ | 𝑌 to  𝑥 ⁢ 𝜎 ^ | 𝑌 ⁢ 𝜎 for each 𝑥 ∈ 𝑋 . Then 𝜎 ¯ is a correctly defined surjective homomorphism.

Proposition 3.7.

Suppose that 𝒞 is a class of groups closed under taking quotient groups, 𝑋  is a group, and  𝑌 is a subgroup of  𝑋 . If  𝑋 is  𝒞 -quasi-regular with respect to  𝑌 and a subgroup 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝑌 ) is normal in  𝑋 , then there exists a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) such that 𝑁 ∩ 𝑌

𝑀 .

Proof.

Suppose that a subgroup 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝑌 ) is normal in  𝑋 . Since the latter is  𝒞 -quasi-regular with respect to  𝑌 , there exists a subgroup 𝑇 ∈ 𝒞 ∗ ⁢ ( 𝑋 ) such that 𝑇 ∩ 𝑌 ⩽ 𝑀 . Let 𝑁

𝑀 ⁢ 𝑇 . Then 𝑁 is normal in  𝑋 and  𝑁 ∩ 𝑌

𝑀 , as is easy to see. Since 𝒞 is closed under taking quotient groups, it follows from the relations 𝑋 / 𝑁 ≅ ( 𝑋 / 𝑇 ) / ( 𝑁 / 𝑇 ) and  𝑋 / 𝑇 ∈ 𝒞 that 𝑋 / 𝑁 ∈ 𝒞 . Thus, 𝑁 is the desired subgroup. ∎

Proposition 3.8.

If 𝒞 is a root class of groups consisting only of periodic groups, then the following statements hold.

1.  Every 𝒞 -group is of finite exponent [52, Proposition 17].

2.  A finite solvable group belongs to  𝒞 if and only if its order is a  𝔓 ⁢ ( 𝒞 ) -number (i.e., each prime divisor of this order lies in  𝔓 ⁢ ( 𝒞 ) ) [35, Proposition 8].

In what follows, the expression

ℙ

⟨ 𝐴 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩

means that ℙ is the generalized free product of groups 𝐴 and  𝐵 with subgroups 𝐻 ⩽ 𝐴 and  𝐾 ⩽ 𝐵 amalgamated by an isomorphism 𝜑 : 𝐻 → 𝐾 . According to [39], subgroups 𝑅 ⩽ 𝐴 and  𝑆 ⩽ 𝐵 are said to be  ( 𝐻 , 𝐾 , 𝜑 ) -compatible if ( 𝑅 ∩ 𝐻 ) ⁢ 𝜑

𝑆 ∩ 𝐾 . Suppose that 𝑅 is normal in  𝐴 , 𝑆  is normal in  𝐵 , and  𝜑 𝑅 , 𝑆 : 𝐻 ⁢ 𝑅 / 𝑅 → 𝐾 ⁢ 𝑆 / 𝑆 is the map taking an element  ℎ ⁢ 𝑅 , ℎ ∈ 𝐻 , to the element  ( ℎ ⁢ 𝜑 ) ⁢ 𝑆 . It follows from the equality ( 𝑅 ∩ 𝐻 ) ⁢ 𝜑

𝑆 ∩ 𝐾 that 𝜑 𝑅 , 𝑆 is a correctly defined isomorphism and, therefore, we can consider the generalized free product

ℙ 𝑅 , 𝑆

⟨ 𝐴 / 𝑅 ∗ 𝐵 / 𝑆 ; 𝐻 𝑅 / 𝑅

𝐾 𝑆 / 𝑆 , 𝜑 𝑅 , 𝑆 ⟩ .

As is easy to see, the identity mapping of the generators of  ℙ into  ℙ 𝑅 , 𝑆 defines a surjective homomorphism 𝜌 𝑅 , 𝑆 : ℙ → ℙ 𝑅 , 𝑆 , whose kernel coincides with the normal closure of the set 𝑅 ∪ 𝑆 in  ℙ . We note also that, if  𝐻 and  𝐾 are normal in  𝐴 and  𝐵 , respectively, then 𝐻 is normal in  ℙ and, therefore, the group  Aut ℙ ⁡ ( 𝐻 ) is defined. Clearly, this group is generated by its subgroups  Aut 𝐴 ⁡ ( 𝐻 ) and  𝜑 ⁢ Aut 𝐵 ⁡ ( 𝐾 ) ⁢ 𝜑 − 1 .

Suppose that 𝑥 ∈ ℙ and  𝑥

𝑥 1 ⁢ 𝑥 2 ⁢ … ⁢ 𝑥 𝑛 , where 𝑛 ⩾ 1 and  𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ∈ 𝐴 ∪ 𝐵 . This product is called a reduced form of  𝑥 if no two adjacent factors  𝑥 𝑖 and  𝑥 𝑖 + 1 lie simultaneously in  𝐴 or  𝐵 . The number  𝑛 is said to be the length of this form. It is known that, if an element 𝑥 ∈ ℙ has at least one reduced form of length greater than  1 , then it is non-trivial (see, e.g., [12, Chapter IV, Theorem 2.6]).

Similar assertions hold for the HNN-extension 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ . A subgroup  𝑄 of  𝐵 is said to be  ( 𝐻 , 𝐾 , 𝜑 ) -compatible if ( 𝑄 ∩ 𝐻 ) ⁢ 𝜑

𝑄 ∩ 𝐾 . When 𝑄 is normal in  𝐵 , this equality ensures that the map 𝜑 𝑄 : 𝐻 ⁢ 𝑄 / 𝑄 → 𝐾 ⁢ 𝑄 / 𝑄 given by the rule ℎ ⁢ 𝑄 ↦ ( ℎ ⁢ 𝜑 ) ⁢ 𝑄 , ℎ ∈ 𝐻 , is a correctly defined isomorphism. Therefore, the HNN-extension

𝔼 𝑄

⟨ 𝐵 / 𝑄 , 𝑡 ; 𝑡 − 1 ( 𝐻 𝑄 / 𝑄 ) 𝑡

𝐾 𝑄 / 𝑄 , 𝜑 𝑄 ⟩

can be considered. As above, the identity mapping of the generators of  𝔼 into  𝔼 𝑄 defines a surjective homomorphism 𝜌 𝑄 : 𝔼 → 𝔼 𝑄 , whose kernel coincides with the normal closure of  𝑄 in  𝔼 .

Obviously, each element 𝑥 ∈ 𝔼 can be represented as a product 𝑥

𝑥 0 ⁢ 𝑡 𝜀 1 ⁢ 𝑥 1 ⁢ … ⁢ 𝑥 𝑛 − 1 ⁢ 𝑡 𝜀 𝑛 ⁢ 𝑥 𝑛 , where 𝑛 ⩾ 0 , 𝑥 0 , 𝑥 1 , … , 𝑥 𝑛 ∈ 𝐵 , and  𝜀 1 , … , 𝜀 𝑛 ∈ { 1 , − 1 } . This product is said to be a reduced form of  𝑥 of length  𝑛 if, for each 𝑖 ∈ { 1 , … , 𝑛 − 1 } , the equalities − 𝜀 𝑖

1

𝜀 𝑖 + 1 imply that 𝑥 𝑖 ∉ 𝐻 , while the equalities 𝜀 𝑖

1

− 𝜀 𝑖 + 1 guarantee that 𝑥 𝑖 ∉ 𝐾 . Britton’s lemma [54] states that, if an element 𝑥 ∈ 𝔼 has a reduced form of non-zero length, then it is non-trivial. The next two propositions are special cases of Theorem 4 from [55] and Theorem 1 from [46].

Proposition 3.9.

Let 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ . If  𝑁 is a normal subgroup of  𝔼 and 𝑁 ∩ 𝐵

1 , then 𝑁 is free.

Proposition 3.10.

Let 𝒞 be a root class of groups. If  𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ , 𝐵  is residually a  𝒞 -group, and there exists a homomorphism of  𝔼 onto a group from  𝒞 acting injectively on  𝐻 and  𝐾 , then 𝔼 is residually a  𝒞 -group.

In what follows, if  𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ , then the expression

𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩

means that 𝔹 is the generalized free product of two isomorphic copies of  𝐵 with the subgroups  𝐻 and  𝐾 amalgamated by the same isomorphism 𝜑 : 𝐻 → 𝐾 . Let 𝜁 𝑔 ⁢ 𝑒 ⁢ 𝑛 be the map of the generators of  𝔹 into  𝔼 given by the rule: 𝑥 ↦ 𝑡 − 1 ⁢ 𝑥 ⁢ 𝑡 , 𝑦 ↦ 𝑦 , where 𝑥 and  𝑦 are generators of the first and second instances of  𝐵 , respectively. Clearly, when extended to a mapping of words, 𝜁 𝑔 ⁢ 𝑒 ⁢ 𝑛  takes all defining relations of  𝔹 to the equalities valid in  𝔼 and, therefore, induces a homomorphism 𝜁 : 𝔹 → 𝔼 . It is also easy to see that, if  𝑥 1 ⁢ … ⁢ 𝑥 𝑛 is a reduced form of an element 𝑥 ∈ 𝔹 ∖ { 1 } , then the product 𝑥 1 ⁢ 𝜁 ⁢ … ⁢ 𝑥 𝑛 ⁢ 𝜁 is a reduced form of the element  𝑥 ⁢ 𝜁 and  𝑥 ⁢ 𝜁 ≠ 1 . Hence, 𝜁  is injective. It can also be noted that, if  𝔼 satisfies  ( ∗ ) , then 𝔘

Aut 𝔹 ⁡ ( 𝐻 ) .

Proposition 3.11.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a class of groups closed under taking subgroups and quotient groups. If there exists a homomorphism  𝜎 of  𝔼 onto a group from  𝒞 acting injectively on  𝐻 and  𝐾 , then 𝔘 , 𝔙 ∈ 𝒞 .

Proof.

It is obvious that 𝔙

Aut 𝔼 ⁡ ( 𝐿 ) . Therefore, the inclusion 𝔙 ∈ 𝒞 follows from Proposition 3.4. Let 𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ , and let 𝜁 : 𝔹 → 𝔼 be the homomorphism defined above. Since 𝜎 is injective on  𝐾

𝐾 ⁢ 𝜁 and  𝒞 is closed under taking subgroups, 𝔹  has a homomorphism onto a group from  𝒞 acting injectively on  𝐻 and  𝐾 . Hence, 𝔘

Aut 𝔹 ⁡ ( 𝐻 ) ∈ 𝒞 by the same Proposition 3.4. ∎

Proposition 3.12.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups closed under taking quotient groups. If  𝔼 is residually a  𝒞 -group, then 𝔘 and  𝔙 are also residually 𝒞 -groups and, therefore, the groups  ℌ ,  𝔎 , 𝔏 , 𝔉 , 𝔘 , and  𝔙 belong to  𝒞 when they are finite.

Proof.

As noted above, the group 𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ can be embedded into the residually 𝒞 -group  𝔼 . Therefore, it is itself residually a  𝒞 -group by Proposition 3.2. Since 𝔘

Aut 𝔹 ⁡ ( 𝐻 ) and  𝔙

Aut 𝔼 ⁡ ( 𝐿 ) , Proposition 3.5 implies that 𝔘 and  𝔙 are also residually 𝒞 -groups. The inclusions ℌ , 𝔎 , 𝔏 , 𝔉 , 𝔘 , 𝔙 ∈ 𝒞 follows from Proposition 3.3. ∎

4.Proof of Theorems 1–2 and Corollary 1 Proposition 4.1.

[53, Theorem 1] If 𝒞 is a root class of groups, ℙ

⟨ 𝐴 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ , 𝐻  is normal in  𝐴 , 𝐾  is normal in  𝐵 , and  𝐴 , 𝐵 , 𝐴 / 𝐻 , 𝐵 / 𝐾 , Aut ℙ ⁡ ( 𝐻 ) ∈ 𝒞 , then there exists a homomorphism of  ℙ onto a group from  𝒞 acting injectively on  𝐴 and  𝐵 .

Let the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfy  ( ∗ ) and  𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ . Then the subgroup  𝐾 of the first free factor of  𝔹 and the subgroup  𝐻 of the second one are  ( 𝐻 , 𝐾 , 𝜑 ) -compatible. Therefore, the generalized free product

𝔹 𝐾 , 𝐻

⟨ ( 𝐵 / 𝐾 ) ∗ ( 𝐵 / 𝐻 ) ; 𝐻 𝐾 / 𝐾

𝐾 𝐻 / 𝐻 , 𝜑 𝐾 , 𝐻 ⟩

and the group  Aut 𝔹 𝐾 , 𝐻 ⁡ ( 𝐻 ⁢ 𝐾 / 𝐾 ) are defined.

Proposition 4.2.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞  is a root class of groups, and  𝐻 ∩ 𝐾

1 . If  𝐵 / 𝐻 , 𝐵 / 𝐾 , 𝐵 / 𝐻 ⁢ 𝐾 , Aut 𝔹 𝐾 , 𝐻 ⁡ ( 𝐻 ⁢ 𝐾 / 𝐾 ) ∈ 𝒞 , then there exists a homomorphism of  𝔼 onto a group from  𝒞 acting injectively on  𝐵 .

Proof.

If 𝒞 contains non-periodic groups, we denote by  ℐ the additive group of the ring  ℤ . Otherwise, let ℐ be the additive group of the ring  ℤ 𝑛 , where 𝑛 is the order of some 𝒞 -group and  𝑛 ⩾ 4 . Since 𝒞 is closed under taking subgroups and extensions, the number  𝑛 with the indicated properties exists and, in both cases, ℐ ∈ 𝒞 .

For each 𝑖 ∈ ℐ , let 𝐵 𝑖 denote an isomorphic copy of  𝐵 . Let also 𝛽 𝑖 : 𝐵 → 𝐵 𝑖 be the corresponding isomorphism, 𝐻 𝑖

𝐻 ⁢ 𝛽 𝑖 , and  𝐾 𝑖

𝐾 ⁢ 𝛽 𝑖 . Consider the group

𝔓

⟨ 𝐵 𝑖 ; 𝐻 𝑖

𝐾 𝑖 − 1 ⁢ ( 𝑖 ∈ ℐ ) ⟩

whose generators are the generators of the groups  𝐵 𝑖 , 𝑖 ∈ ℐ , and whose defining relations are those of  𝐵 𝑖 , 𝑖 ∈ ℐ , and all possible relations of the form ℎ ⁢ 𝜑 ⁢ 𝛽 𝑖 − 1

ℎ ⁢ 𝛽 𝑖 , where ℎ ∈ 𝐻 and  𝑖 ∈ ℐ . It is easy to see that, if  ℐ is infinite, then 𝔓 is the tree product of the groups  𝐵 𝑖 , 𝑖 ∈ ℐ , that corresponds to an infinite chain. Otherwise, 𝔓  is the polygonal product of the same groups. Theorem 1 from [56] says that, in the first case, the identity mappings of the generators of  𝐵 𝑖 , 𝑖 ∈ ℐ , into  𝔓 can be extended to injective homomorphisms. It follows from the relations 𝐻 ∩ 𝐾

1 and  𝑛 ⩾ 4 that the same statement holds in the second case [57].

Let 𝛼 𝑔 ⁢ 𝑒 ⁢ 𝑛 be the mapping of the generators of  𝔓 acting as the isomorphisms  𝛽 𝑖 − 1 ⁢ 𝛽 𝑖 + 1 , 𝑖 ∈ ℐ . Clearly, 𝛼 𝑔 ⁢ 𝑒 ⁢ 𝑛  defines an automorphism  𝛼 of  𝔓 , whose order is equal to the order of  ℐ . Let 𝔔 denote the splitting extension of  𝔓 by the cyclic group  ⟨ 𝛼 ⟩ . Consider the mapping  𝜆 𝑔 ⁢ 𝑒 ⁢ 𝑛 of the generators of  𝔼 into  𝔔 that acts on the generators of  𝐵 as  𝛽 0 and takes  𝑡 to  𝛼 (here and below, we identify the groups  𝐵 , 𝐵 𝑖 , 𝑖 ∈ ℐ , and  𝔓 with the corresponding subgroups of  𝔼 , 𝔓 , and  𝔔 , respectively). It is easy to see that, when extended to a mapping of words, 𝜆 𝑔 ⁢ 𝑒 ⁢ 𝑛  takes all defining relations of  𝔼 to the equalities valid in  𝔔 . Therefore, it induces a homomorphism 𝜆 : 𝔼 → 𝔔 , which acts on  𝐵 as  𝛽 0 . Since 𝔔 is obviously generated by the set { 𝛼 } ∪ { 𝑏 ⁢ 𝛽 0 ∣ 𝑏 ∈ 𝐵 } , the homomorphism  𝜆 is surjective.

Let 𝑖 ∈ ℐ . It is clear that 𝛽 𝑖 + 1 and  𝛽 𝑖 induce an isomorphism  𝛾 𝑖 of  𝔹 𝐾 , 𝐻 onto the generalized free product

𝔓 𝑖

⟨ ( 𝐵 𝑖 + 1 / 𝐾 𝑖 + 1 ) ∗ ( 𝐵 𝑖 / 𝐻 𝑖 ) ; 𝐻 𝑖 + 1 𝐾 𝑖 + 1 / 𝐾 𝑖 + 1

𝐾 𝑖 𝐻 𝑖 / 𝐻 𝑖 , 𝜑 𝑖 ⟩ ,

where the isomorphism 𝜑 𝑖 : 𝐻 𝑖 + 1 ⁢ 𝐾 𝑖 + 1 / 𝐾 𝑖 + 1 → 𝐾 𝑖 ⁢ 𝐻 𝑖 / 𝐻 𝑖 is given by the rule:

( ℎ ⁢ 𝛽 𝑖 + 1 ) ⁢ 𝐾 𝑖 + 1 ↦ ( ℎ ⁢ 𝜑 ⁢ 𝛽 𝑖 ) ⁢ 𝐻 𝑖 , ℎ ∈ 𝐻 .

The relations

( 𝐵 / 𝐻 ) / ( 𝐾 ⁢ 𝐻 / 𝐻 ) ≅ 𝐵 / 𝐻 ⁢ 𝐾 ≅ ( 𝐵 / 𝐾 ) / ( 𝐻 ⁢ 𝐾 / 𝐾 )

and  𝐵 / 𝐻 ⁢ 𝐾 ∈ 𝒞 mean that all conditions of Proposition 4.1 hold for the group  𝔹 𝐾 , 𝐻 . Since ( 𝐵 / 𝐾 ) ⁢ 𝛾 𝑖

𝐵 𝑖 + 1 / 𝐾 𝑖 + 1 and  ( 𝐵 / 𝐻 ) ⁢ 𝛾 𝑖

𝐵 𝑖 / 𝐻 𝑖 , it follows that there exists a homomorphism  𝜂 𝑖 of  𝔓 𝑖 onto a group from  𝒞 satisfying the equalities

ker ⁡ 𝜂 𝑖 ∩ 𝐵 𝑖 / 𝐻 𝑖

1

ker ⁡ 𝜂 𝑖 ∩ 𝐵 𝑖 + 1 / 𝐾 𝑖 + 1 .

Consider the mapping of the generators of  𝔓 into  𝔓 𝑖 which acts identically on the elements of  𝐵 𝑖 and  𝐵 𝑖 + 1 , and takes the generators of other free factors to  1 . It is easy to see that this mapping defines a surjective homomorphism 𝜃 𝑖 : 𝔓 → 𝔓 𝑖 . Therefore, if  𝑀 𝑖 denotes the subgroup ker ⁡ 𝜃 𝑖 ⁢ 𝜂 𝑖 , we have 𝑀 𝑖 ∈ 𝒞 ∗ ⁢ ( 𝔓 ) , 𝑀 𝑖 ∩ 𝐵 𝑖

𝐻 𝑖 , and  𝑀 𝑖 ∩ 𝐵 𝑖 + 1

𝐾 𝑖 + 1 .

Let 𝑀

𝑀 0 ∩ 𝑀 − 1 . It follows from Proposition 3.3 and the above that 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝔓 )  and

𝑀 ∩ 𝐵 0

( 𝑀 0 ∩ 𝐵 0 ) ∩ ( 𝑀 − 1 ∩ 𝐵 0 )

𝐻 0 ∩ 𝐾 0

( 𝐻 ∩ 𝐾 ) ⁢ 𝛽 0

1 .

Since ⟨ 𝛼 ⟩ ≅ ℐ ∈ 𝒞 , the factors of the subnormal sequence 𝑀 ⩽ 𝔓 ⩽ 𝔔 belong to  𝒞 . Hence, by Gruenberg’s condition, there exists a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔔 ) lying in  𝑀 . It now follows from the relations 𝐵 0

𝐵 ⁢ 𝜆 and  𝑁 ∩ 𝐵 0 ⩽ 𝑀 ∩ 𝐵 0

1 that the composition of  𝜆 and the natural homomorphism 𝔔 → 𝔔 / 𝑁 is the desired mapping. ∎

Proposition 4.3.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups. If  𝐿 , 𝐵 / 𝐻 , 𝐵 / 𝐾 , 𝐵 / 𝐻 ⁢ 𝐾 , Aut 𝔼 ⁡ ( 𝐿 ) , Aut 𝔹 𝐾 , 𝐻 ⁡ ( 𝐻 ⁢ 𝐾 / 𝐾 ) ∈ 𝒞 , then there exists a homomorphism of  𝔼 onto a group from  𝒞 acting injectively on  𝐵 .

Proof.

Since 𝐿 is a normal subgroup of  𝐵 and  ( 𝐿 ∩ 𝐻 ) ⁢ 𝜑

𝐿 ⁢ 𝜑

𝐿

𝐿 ∩ 𝐾 , the HNN-extension

𝔼 𝐿

⟨ 𝐵 / 𝐿 , 𝑡 ; 𝑡 − 1 ( 𝐻 / 𝐿 ) 𝑡

𝐾 / 𝐿 , 𝜑 𝐿 ⟩

is defined. Consider the following groups:

𝔹 𝐾 , 𝐻

⟨ ( 𝐵 / 𝐾 ) ∗ ( 𝐵 / 𝐻 ) ; 𝐻 𝐾 / 𝐾

𝐾 𝐻 / 𝐻 , 𝜑 𝐾 , 𝐻 ⟩ ,

𝔹 𝐾 / 𝐿 , 𝐻 / 𝐿

⟨ ( 𝐵 / 𝐿 ) / ( 𝐾 / 𝐿 ) ∗ ( 𝐵 / 𝐿 ) / ( 𝐻 / 𝐿 ) ;

( 𝐻 / 𝐿 ) ( 𝐾 / 𝐿 ) / ( 𝐾 / 𝐿 )

( 𝐾 / 𝐿 ) ( 𝐻 / 𝐿 ) / ( 𝐻 / 𝐿 ) , 𝜑 𝐾 / 𝐿 , 𝐻 / 𝐿 ⟩ .

It is clear that the identity mapping of the generators of  𝔹 𝐾 / 𝐿 , 𝐻 / 𝐿 into  𝔹 𝐾 , 𝐻 defines an isomorphism, which takes the subgroup ( 𝐻 / 𝐿 ) ⁢ ( 𝐾 / 𝐿 ) / ( 𝐾 / 𝐿 ) onto  𝐻 ⁢ 𝐾 / 𝐾 . Therefore,

Aut 𝔹 𝐾 / 𝐿 , 𝐻 / 𝐿 ⁡ ( ( 𝐻 / 𝐿 ) ⁢ ( 𝐾 / 𝐿 ) / ( 𝐾 / 𝐿 ) ) ≅ Aut 𝔹 𝐾 , 𝐻 ⁡ ( 𝐻 ⁢ 𝐾 / 𝐾 ) ∈ 𝒞 .

Since

( 𝐵 / 𝐿 ) / ( 𝐻 / 𝐿 ) ≅ 𝐵 / 𝐻 ∈ 𝒞 , ( 𝐵 / 𝐿 ) / ( 𝐾 / 𝐿 ) ≅ 𝐵 / 𝐾 ∈ 𝒞 ,

( 𝐵 / 𝐿 ) / ( 𝐻 / 𝐿 ) ⁢ ( 𝐾 / 𝐿 ) ≅ 𝐵 / 𝐻 ⁢ 𝐾 ∈ 𝒞 , 𝐻 / 𝐿 ∩ 𝐾 / 𝐿

1 ,

the HNN-extension 𝔼 𝐿 satisfies the conditions of Proposition 4.2. Hence, there exists a homomorphism 𝜏 𝐿 of  𝔼 𝐿 onto a group from  𝒞 which is injective on  𝐵 / 𝐿 . By Proposition 3.9, the kernel of  𝜏 𝐿 is a free group.

Since 𝐿 is normal in  𝔼 , the equality 𝐿

ker ⁡ 𝜌 𝐿 holds. Therefore, the subgroup 𝑈

ker ⁡ 𝜌 𝐿 ⁢ 𝜏 𝐿 is an extension of  𝐿 by a free group. It is well known that such an extension is splittable, i.e.,  𝑈 has a free subgroup  𝐹 satisfying the relations 𝑈

𝐿 ⁢ 𝐹 and  𝐿 ∩ 𝐹

1 .

Let 𝜉 : 𝔼 → Aut ⁡ 𝐿 be the homomorphism taking an element 𝑥 ∈ 𝔼 to the automorphism  𝑥 ^ | 𝐿 . Its kernel obviously coincides with the centralizer  𝒵 𝔼 ⁢ ( 𝐿 ) of  𝐿 in  𝔼 . Since 𝒞 is closed under taking subgroups and extensions, it follows from this fact and the relations

𝑈

𝐿 ⁢ 𝐹 , 𝐿 ∩ 𝐹

1 , 𝐿 ⩽ 𝐵 , 𝐹 ⁢ 𝒵 𝔼 ⁢ ( 𝐿 ) / 𝒵 𝔼 ⁢ ( 𝐿 ) ⩽ 𝔼 / 𝒵 𝔼 ⁢ ( 𝐿 ) , Im ⁡ 𝜉

Aut 𝔼 ⁡ ( 𝐿 ) ∈ 𝒞

that the subgroup 𝑉

𝒵 𝔼 ⁢ ( 𝐿 ) ∩ 𝐹 is normal in  𝑈 ,

𝐹 / 𝑉 ≅ 𝐹 ⁢ 𝒵 𝔼 ⁢ ( 𝐿 ) / 𝒵 𝔼 ⁢ ( 𝐿 ) ∈ 𝒞 , 𝐿 ⁢ 𝑉 / 𝑉 ≅ 𝐿 / 𝐿 ∩ 𝑉 ≅ 𝐿 ∈ 𝒞 ,

𝑈 / 𝐿 ⁢ 𝑉

𝐿 ⁢ 𝐹 / 𝐿 ⁢ 𝑉 ≅ 𝐹 / 𝑉 ⁢ ( 𝐿 ∩ 𝐹 )

𝐹 / 𝑉 ∈ 𝒞 ,

and the quotient group  𝑈 / 𝑉 belongs to  𝒞 as an extension of  𝐿 ⁢ 𝑉 / 𝑉 by a group isomorphic to  𝑈 / 𝐿 ⁢ 𝑉 . In addition, 𝑈 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) due to the definition of  𝜏 𝐿 . Thus, we can apply Gruenberg’s condition to the subnormal series 1 ⩽ 𝑉 ⩽ 𝑈 ⩽ 𝔼 and find a subgroup 𝑊 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) lying in  𝑉 .

Since 𝜏 𝐿 acts injectively on  𝐵 / 𝐿 , the equality 𝑈 ∩ 𝐵

𝐿 holds. It now follows from the inclusions 𝑊 ⩽ 𝑉 ⩽ 𝐹 ⩽ 𝑈 that 𝑊 ∩ 𝐵 ⩽ 𝐹 ∩ ( 𝑈 ∩ 𝐵 )

𝐹 ∩ 𝐿

1 . Hence, the natural homomorphism 𝔼 → 𝔼 / 𝑊 is the desired one. ∎

Proof of Theorem 1.

The implication 1 ⇒ 2 and the residual 𝒞 -ness of  𝔼 follow from Propositions 3.11 and 3.10, respectively. To prove the implication 2 ⇒ 1 , it is sufficient to show that all conditions of Proposition 4.3 hold if 𝔘 , 𝔙 ∈ 𝒞 .

Indeed, since 𝒞 is closed under taking subgroups and quotient groups, the inclusions 𝐿 , 𝐵 / 𝐻 , 𝐵 / 𝐾 , 𝐵 / 𝐻 ⁢ 𝐾 ∈ 𝒞 follow from the condition 𝐵 ∈ 𝒞 . Let 𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ . By Proposition 3.6, the relations Aut 𝔹 ⁡ ( 𝐻 )

𝔘 ∈ 𝒞 , 𝔹 𝐾 , 𝐻

𝔹 ⁢ 𝜌 𝐾 , 𝐻 , and  𝐻 ⁢ 𝐾 / 𝐾

𝐻 ⁢ 𝜌 𝐾 , 𝐻 imply that Aut 𝔹 𝐾 , 𝐻 ⁡ ( 𝐻 ⁢ 𝐾 / 𝐾 ) ∈ 𝒞 . It remains to note that Aut 𝔼 ⁡ ( 𝐿 )

𝔙 ∈ 𝒞 . ∎

Proposition 4.4.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝑄  is a normal ( 𝐻 , 𝐾 , 𝜑 ) -compatible subgroup of  𝐵 , and

𝔼 𝑄

⟨ 𝐵 / 𝑄 , 𝑡 ; 𝑡 − 1 ( 𝐻 𝑄 / 𝑄 ) 𝑡

𝐾 𝑄 / 𝑄 , 𝜑 𝑄 ⟩ .

Suppose also that the symbols  ℌ 𝑄 , 𝔎 𝑄 , and  𝔏 𝑄 denote the subgroups

Aut 𝐵 / 𝑄 ⁡ ( 𝐻 ⁢ 𝑄 / 𝑄 ) , 𝜑 𝑄 ⁢ Aut 𝐵 / 𝑄 ⁡ ( 𝐾 ⁢ 𝑄 / 𝑄 ) ⁢ 𝜑 𝑄 − 1 , and Aut 𝐵 / 𝑄 ⁡ ( 𝐿 ⁢ 𝑄 / 𝑄 ) ,

respectively. Then the following statements hold.

1.  There exists a homomorphism of  𝔘 onto the group 𝔘 𝑄

sgp ⁡ { ℌ 𝑄 , 𝔎 𝑄 } which maps the subgroups  ℌ and  𝔎 onto  ℌ 𝑄 and  𝔎 𝑄 , respectively.

2.  The subgroup  𝐿 ⁢ 𝑄 / 𝑄 is  𝜑 𝑄 -invariant and there exists a homomorphism of  𝔙 onto the group 𝔙 𝑄

sgp ⁡ { 𝔏 𝑄 , 𝜑 𝑄 | 𝐿 ⁢ 𝑄 / 𝑄 } which maps 𝔏 and  𝔉 onto the subgroups  𝔏 𝑄 and 𝔉 𝑄

⟨ 𝜑 𝑄 | 𝐿 ⁢ 𝑄 / 𝑄 ⟩ , respectively.

Proof.

1.  Let 𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ . Since 𝑄 is  ( 𝐻 , 𝐾 , 𝜑 ) -compatible, the generalized free product

𝔹 𝑄 , 𝑄

⟨ 𝐵 / 𝑄 ∗ 𝐵 / 𝑄 ; 𝐻 𝑄 / 𝑄

𝐾 𝑄 / 𝑄 , 𝜑 𝑄 , 𝑄 ⟩

is defined. It follows from Proposition 3.6 that the map 𝜌 𝑄 , 𝑄 ¯ : Aut 𝔹 ⁡ ( 𝐻 ) → Aut 𝔹 𝑄 , 𝑄 ⁡ ( 𝐻 ⁢ 𝜌 𝑄 , 𝑄 ) given by the rule 𝑥 ^ | 𝐻 ↦ 𝑥 ⁢ 𝜌 𝑄 , 𝑄 ^ | 𝐻 ⁢ 𝜌 𝑄 , 𝑄 , 𝑥 ∈ 𝔹 , is a surjective homomorphism. Clearly, 𝔘 𝑄

Aut 𝔹 𝑄 , 𝑄 ⁡ ( 𝐻 ⁢ 𝑄 / 𝑄 ) , 𝔘

Aut 𝔹 ⁡ ( 𝐻 ) , 𝐻 ⁢ 𝜌 𝑄 , 𝑄

𝐻 ⁢ 𝑄 / 𝑄 , ℌ ⁢ 𝜌 𝑄 , 𝑄 ¯

ℌ 𝑄 , and  𝔎 ⁢ 𝜌 𝑄 , 𝑄 ¯

𝔎 𝑄 . Therefore, the homomorphism 𝜌 𝑄 , 𝑄 ¯ is desired.

2.  The equality 𝐿 ⁢ 𝜑

𝐿 and the definition of  𝜑 𝑄 imply that ( 𝐿 ⁢ 𝑄 / 𝑄 ) ⁢ 𝜑 𝑄

𝐿 ⁢ 𝑄 / 𝑄 . Since 𝜑 𝑄 is an isomorphism, this relation means that 𝜑 𝑄 | 𝐿 ⁢ 𝑄 / 𝑄 ∈ Aut ⁡ 𝐿 ⁢ 𝑄 / 𝑄 . The existence of the desired homomorphism is ensured by Proposition 3.6 due to the equalities 𝔙 𝑄

Aut 𝔼 𝑄 ⁡ ( 𝐿 ⁢ 𝑄 / 𝑄 ) , 𝔙

Aut 𝔼 ⁡ ( 𝐿 ) , and  𝐿 ⁢ 𝑄 / 𝑄

𝐿 ⁢ 𝜌 𝑄 . ∎

Proof of Theorem 2.

Statement 2 follows from Statement 1 and Proposition 3.10. Let us prove Statement 1. If there exists a homomorphism of  𝔼 onto a group from  𝒞 that extends  𝜎 , then 𝔘 , 𝔙 ∈ 𝒞 by Proposition 3.11. It remains to show that the converse also holds.

Let 𝑄

ker ⁡ 𝜎 . Then 𝐵 / 𝑄 ∈ 𝒞 and it follows from the equality 𝑄 ∩ 𝐻 ⁢ 𝐾

1 that 𝑄 ∩ 𝐻

1

𝑄 ∩ 𝐾 and  𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄

𝐿 ⁢ 𝑄 / 𝑄 . Hence, the groups  𝔼 𝑄 , 𝔘 𝑄 and  𝔙 𝑄 can be defined as in Proposition 4.4. By this proposition, the subgroup  𝐿 ⁢ 𝑄 / 𝑄 is  𝜑 𝑄 -invariant. Since 𝔘 , 𝔙 ∈ 𝒞 and  𝒞 is closed under taking quotient groups, Proposition 4.4 also implies that 𝔘 𝑄 , 𝔙 𝑄 ∈ 𝒞 . By Theorem 1, it follows from these relations and the equality 𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄

𝐿 ⁢ 𝑄 / 𝑄 that there exists a homomorphism  𝜏 of the group  𝔼 𝑄 which satisfies the conditions ker ⁡ 𝜏 ∩ 𝐵 / 𝑄

1 and  Im ⁡ 𝜏 ∈ 𝒞 . Since 𝜌 𝑄 extends  𝜎 , the composition  𝜌 𝑄 ⁢ 𝜏 is the desired mapping. ∎

Proof of Corollary 1.

Necessity. Proposition 3.3 states that there exists a homomorphism of  𝔼 onto a group from  𝒞 acting injectively on the finite subgroup  𝐻 ⁢ 𝐾 . Hence, 𝔘 , 𝔙 ∈ 𝒞 by Proposition 3.11. The residual 𝒞 -ness of  𝐵 is ensured by Proposition 3.2.

Sufficiency. As above, if  𝐵 is residually a  𝒞 -group, it has a homomorphism onto a group from  𝒞 acting injectively on  𝐻 ⁢ 𝐾 . Therefore, we can use Statement 2 of Theorem 2 to prove the residual 𝒞 -ness of  𝔼 . ∎

5.Proof of Theorem 3 Proposition 5.1.

[52, Proposition 9] Let ℙ

⟨ 𝐴 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ . Suppose also that 𝐻 is normal in  𝐴 , 𝐾 is normal in  𝐵 , and the group  Aut ℙ ⁡ ( 𝐻 ) coincides with one of its subgroups  Aut 𝐴 ⁡ ( 𝐻 ) and  𝜑 ⁢ Aut 𝐵 ⁡ ( 𝐻 ) ⁢ 𝜑 − 1 . If  𝒞 is a class of groups closed under taking subgroups and  ℙ is residually a  𝒞 -group, then the following statements hold.

1.  If 𝐾 ≠ 𝐵 , then 𝐻 is  𝒞 -separable in  𝐴 .

2.  If 𝐻 ≠ 𝐴 , then 𝐾 is  𝒞 -separable in  𝐵 .

Proposition 5.2.

[47, Theorem 1] Let 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ . Suppose also that 𝒞 is a class of groups, the symbols  𝐻 ¯ and  𝐾 ¯ denote the subgroups ⋂ 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) 𝐻 ⁢ ( 𝑁 ∩ 𝐵 ) and  ⋂ 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) 𝐾 ⁢ ( 𝑁 ∩ 𝐵 ) , respectively, and at least one of the following statements holds:

( 𝑎 )

the subgroups  𝐻 and  𝐾 coincide and satisfy a non-trivial identity;

( 𝑏 )

the subgroups  𝐻 and  𝐾 are properly contained in a subgroup  𝐷 of  𝐵 satisfying a non-trivial identity.

If 𝔼 is residually a  𝒞 -group, then 𝐻 ¯

𝐻 and  𝐾 ¯

𝐾 .

Proof of Theorem 3.

Suppose that the subgroups  𝐻 ¯ and  𝐾 ¯ are defined as in Proposition 5.2. Since ( 𝐻 ∩ 𝐾 ) ⁢ 𝜑

𝐻 ∩ 𝐾 , the relations 𝐻 ⩽ 𝐾 , 𝐻

𝐾 , and  𝐻 ⩾ 𝐾 are equivalent. Therefore, only two cases are possible: 𝐻

𝐾 and  𝐻 ≠ 𝐻 ⁢ 𝐾 ≠ 𝐾 . If  𝐻

𝐵

𝐾 , then the residual 𝒞 -ness of  𝐵 / 𝐻 and  𝐵 / 𝐾 is obvious. Hence, we can further assume that 𝐻 ≠ 𝐵 ≠ 𝐾 .

Due to Proposition 3.1, to end the proof it suffices to show that 𝐻 and  𝐾 are  𝒞 -separable in  𝐵 . If  𝔘

ℌ or  𝔘

𝔎 , then the group 𝔹

⟨ 𝐵 ∗ 𝐵 ; 𝐻

𝐾 , 𝜑 ⟩ satisfies all conditions of Proposition 5.1, which ensures the required separability.

Suppose that 𝐻 and  𝐾 satisfy a non-trivial identity. Then the group  𝐻 ⁢ 𝐾 has the same property since it is an extension of  𝐾 by a group isomorphic to  𝐻 ⁢ 𝐾 / 𝐾 and  𝐻 ⁢ 𝐾 / 𝐾 ≅ 𝐻 / 𝐻 ∩ 𝐾 . Thus, the conditions of Proposition 5.2 hold, and we get the equalities 𝐻 ¯

𝐻 and  𝐾 ¯

𝐾 . It remains to note that, if  𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) , then 𝑁 ∩ 𝐵 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) by Proposition 3.2, and therefore these equalities imply the  𝒞 -separability of  𝐻 and  𝐾 in  𝐵 .

Now suppose that ℌ satisfies a non-trivial identity. Then, by Lemma 2 from [15], it satisfies a non-trivial identity of the form

𝜔 ⁢ ( 𝑦 , 𝑥 1 , 𝑥 2 )

𝜔 0 ⁢ ( 𝑥 1 , 𝑥 2 ) ⁢ 𝑦 𝜀 1 ⁢ 𝜔 1 ⁢ ( 𝑥 1 , 𝑥 2 ) ⁢ … ⁢ 𝑦 𝜀 𝑛 ⁢ 𝜔 𝑛 ⁢ ( 𝑥 1 , 𝑥 2 ) ,

where 𝑛 ⩾ 1 , 𝜀 1 , … , 𝜀 𝑛

± 1 , and  𝜔 0 ⁢ ( 𝑥 1 , 𝑥 2 ) , … , 𝜔 𝑛 ⁢ ( 𝑥 1 , 𝑥 2 ) ∈ { 𝑥 1 ± 1 , 𝑥 2 ± 1 , ( 𝑥 1 ⁢ 𝑥 2 − 1 ) ± 1 } . Let us assume that there exist elements 𝑢 1 , 𝑢 2 , 𝑣 ∈ 𝔼 with the following properties:

( 𝔦 )

the commutator [ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of non-zero length;

( 𝔦 ⁢ 𝔦 )

for each subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) , the inclusions 𝑢 1 , 𝑢 2 ∈ 𝐵 ⁢ 𝑁 and  𝑣 ∈ 𝐻 ⁢ 𝑁 hold.

Then 𝔼 is not residually a  𝒞 -group, contrary to the condition of the theorem.

Indeed, since the element 𝑔

[ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of non-zero length, it is not equal to  1 . At the same time, if  𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) , then 𝑔 ≡ [ 𝜔 ⁢ ( ℎ , 𝑏 1 , 𝑏 2 ) , ℎ ] ( mod 𝑁 ) for some 𝑏 1 , 𝑏 2 ∈ 𝐵 and  ℎ ∈ 𝐻 . The restriction to  𝐻 of the conjugation by  𝜔 ⁢ ( ℎ , 𝑏 1 , 𝑏 2 ) coincides with the element 𝜔 ⁢ ( ℎ ^ , 𝑏 1 ^ | 𝐻 , 𝑏 2 ^ | 𝐻 ) of  ℌ , which is the identity mapping because ℌ satisfies  𝜔 . Therefore, [ 𝜔 ⁢ ( ℎ , 𝑏 1 , 𝑏 2 ) , ℎ ]

1 , 𝑔 ≡ 1 ( mod 𝑁 ) , and  𝔼 is not residually a  𝒞 -group since 𝑁 is chosen arbitrarily.

As above, to prove the  𝒞 -separability of  𝐻 and  𝐾 in  𝐵 , it suffices to show that 𝐻 ¯

𝐻 and  𝐾 ¯

𝐾 . Arguing by contradiction, we consider four cases and, in each of them, find elements 𝑢 1 , 𝑢 2 , 𝑣 ∈ 𝔼 satisfying  ( 𝔦 ) and  ( 𝔦 ⁢ 𝔦 ) .

Case 1. 𝐻 ¯ ≠ 𝐻 and  [ 𝐵 : 𝐻 ] ⩾ 3 .

Let 𝑏 1 ∈ 𝐻 ¯ ∖ 𝐻 . Since [ 𝐵 : 𝐻 ] ⩾ 3 and  𝐾 ≠ 𝐵 , there exist elements 𝑏 2 , 𝑐 ∈ 𝐵 such that 𝑏 2 , 𝑏 1 ⁢ 𝑏 2 − 1 ∉ 𝐻 and  𝑐 ∉ 𝐾 . Let us put 𝑢 1

𝑏 1 , 𝑢 2

𝑏 2 , and  𝑣

𝑡 ⁢ 𝑐 − 1 ⁢ 𝑡 − 1 ⁢ 𝑏 1 ⁢ 𝑡 ⁢ 𝑐 ⁢ 𝑡 − 1 .

Since 𝑏 1 ± 1 , 𝑏 2 ± 1 , ( 𝑏 1 ⁢ 𝑏 2 − 1 ) ± 1 ∉ 𝐻 and  𝑐 ∉ 𝐾 , the element [ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of length  8 ⁢ ( 𝑛 + 1 ) . At the same time, if  𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) , then it follows from the inclusion 𝑏 1 ∈ 𝐻 ¯ that 𝑏 1 ∈ 𝐻 ⁢ 𝑁 , 𝑡 − 1 ⁢ 𝑏 1 ⁢ 𝑡 ∈ 𝐾 ⁢ 𝑁 , 𝑐 − 1 ⁢ 𝑡 − 1 ⁢ 𝑏 1 ⁢ 𝑡 ⁢ 𝑐 ∈ 𝐾 ⁢ 𝑁 (because 𝐾 is normal in  𝐵 ), and  𝑡 ⁢ 𝑐 − 1 ⁢ 𝑡 − 1 ⁢ 𝑏 1 ⁢ 𝑡 ⁢ 𝑐 ⁢ 𝑡 − 1 ∈ 𝐻 ⁢ 𝑁 . Thus, the elements  𝑢 1 ,  𝑢 2 , and  𝑣 satisfy  ( 𝔦 ) and  ( 𝔦 ⁢ 𝔦 ) .

Case 2. 𝐻 ¯ ≠ 𝐻 and  [ 𝐵 : 𝐻 ]

2 .

Let us fix some elements 𝑏 ∈ 𝐻 ¯ ∖ 𝐻 and  𝑐 ∈ 𝐵 ∖ 𝐾 , and put 𝑢 1

𝑡 − 1 ⁢ 𝑏 ⁢ 𝑡 , 𝑢 2

𝑡 − 2 ⁢ 𝑏 ⁢ 𝑡 2 , and  𝑣

𝑐 . Then 𝑢 1 ⁢ 𝑢 2 − 1

𝑡 − 1 ⁢ 𝑏 ⁢ 𝑡 − 1 ⁢ 𝑏 − 1 ⁢ 𝑡 2 and the element [ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of length not less than  4 ⁢ ( 𝑛 + 1 ) . The relations [ 𝐵 : 𝐻 ]

2 and  𝐻 ¯ ≠ 𝐻 mean that 𝐵

𝐻 ¯ . Therefore, 𝑏 ∈ 𝐻 ⁢ 𝑁 and  𝐵 ⁢ 𝑁

𝐻 ⁢ 𝑁 for any 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) . It follows that

𝑢 1

𝑡 − 1 ⁢ 𝑏 ⁢ 𝑡 ∈ 𝐾 ⁢ 𝑁 ⩽ 𝐵 ⁢ 𝑁

𝐻 ⁢ 𝑁 , 𝑢 2

𝑡 − 2 ⁢ 𝑏 ⁢ 𝑡 2

𝑡 − 1 ⁢ 𝑢 1 ⁢ 𝑡 ∈ 𝐾 ⁢ 𝑁 ,

and  𝑣

𝑐 ∈ 𝐵 ⁢ 𝑁

𝐻 ⁢ 𝑁 , as required.

Case 3. 𝐾 ¯ ≠ 𝐾 and  [ 𝐵 : 𝐻 ] ⩾ 3 .

Suppose that 𝑐 ∈ 𝐾 ¯ ∖ 𝐾 and elements 𝑏 1 , 𝑏 2 ∈ 𝐵 ∖ 𝐻 are such that 𝑏 1 ⁢ 𝑏 2 − 1 ∉ 𝐻 . Let us put 𝑢 1

𝑏 1 , 𝑢 2

𝑏 2 , and  𝑣

𝑡 ⁢ 𝑐 ⁢ 𝑡 − 1 . Then the element [ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of length  4 ⁢ ( 𝑛 + 1 ) and, for each subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) , the inclusions 𝑏 1 , 𝑏 2 ∈ 𝐵 ⁢ 𝑁 , 𝑐 ∈ 𝐾 ⁢ 𝑁 , and  𝑡 ⁢ 𝑐 ⁢ 𝑡 − 1 ∈ 𝐻 ⁢ 𝑁 hold.

Case 4. 𝐾 ¯ ≠ 𝐾 and  [ 𝐵 : 𝐻 ]

2 .

It follows from the relations 𝐾 / 𝐿

𝐾 / 𝐻 ∩ 𝐾 ≅ 𝐾 ⁢ 𝐻 / 𝐻 ⩽ 𝐵 / 𝐻 that [ 𝐾 : 𝐿 ] ⩽ 2 . If  𝐾

𝐿 , then 𝐾 ⩽ 𝐻 and, hence, 𝐾

𝐻 , as noted at the beginning of the proof. The last equality means that 𝐻 ¯

𝐾 ¯ and  𝐻 ¯ ≠ 𝐻 , which is impossible due to Cases 1 and 2 considered above. Thus, [ 𝐾 : 𝐿 ]

2 . Let us put 𝐿 ¯

⋂ 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) 𝐿 ⁢ ( 𝑁 ∩ 𝐵 ) and show that 𝐿 ¯ ≠ 𝐿 .

Suppose, on the contrary, that 𝐿 ¯

𝐿 , and fix some elements 𝑐 ∈ 𝐾 ¯ ∖ 𝐾 and  𝑘 ∈ 𝐾 ∖ 𝐿 . Since [ 𝐾 : 𝐿 ]

2 and  𝑐 ∉ 𝐾 , the relations 𝐾

𝐿 ∪ 𝑘 ⁢ 𝐿 and  𝑐 , 𝑘 − 1 ⁢ 𝑐 ∉ 𝐿

𝐿 ¯ hold. Hence, 𝑐 ∉ 𝐿 ⁢ ( 𝑁 1 ∩ 𝐵 ) and  𝑘 − 1 ⁢ 𝑐 ∉ 𝐿 ⁢ ( 𝑁 2 ∩ 𝐵 ) for suitable subgroups 𝑁 1 , 𝑁 2 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) . Let  𝑁

𝑁 1 ∩ 𝑁 2 . Then 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) by Proposition 3.3 and  𝑐 , 𝑘 − 1 ⁢ 𝑐 ∉ 𝐿 ⁢ ( 𝑁 ∩ 𝐵 ) . It follows that

𝑐 ∉ 𝐿 ⁢ ( 𝑁 ∩ 𝐵 ) ∪ 𝑘 ⁢ 𝐿 ⁢ ( 𝑁 ∩ 𝐵 )

𝐾 ⁢ ( 𝑁 ∩ 𝐵 )

and, therefore, 𝑐 ∉ 𝐾 ¯ despite the choice of  𝑐 .

So, 𝐿 ¯ ≠ 𝐿 . Let us fix some elements 𝑏 ∈ 𝐿 ¯ ∖ 𝐿 and  𝑑 ∈ 𝐵 ∖ 𝐻 , and put 𝑢 1

𝑡 ⁢ 𝑏 ⁢ 𝑡 − 1 , 𝑢 2

𝑡 2 ⁢ 𝑏 ⁢ 𝑡 − 2 , and  𝑣

𝑑 ⁢ 𝑡 ⁢ 𝑏 ⁢ 𝑡 − 1 ⁢ 𝑑 − 1 . Then 𝑢 1 ⁢ 𝑢 2 − 1

𝑡 ⁢ 𝑏 ⁢ 𝑡 ⁢ 𝑏 − 1 ⁢ 𝑡 − 2 . Since 𝐿 ⩽ 𝐻 , the relations 𝐿 ¯ ⩽ 𝐻 ¯

𝐻 and  𝑏 ∈ 𝐻 hold. Therefore, 𝑏 ∉ 𝐾 and the element [ 𝜔 ⁢ ( 𝑣 , 𝑢 1 , 𝑢 2 ) , 𝑣 ] has a reduced form of length not less than  8 ⁢ ( 𝑛 + 1 ) . At the same time, since 𝑏 ∈ 𝐿 ¯ and  𝐿 is normal in  𝔼 , the inclusions 𝑏 ∈ 𝐿 ⁢ 𝑁 , 𝑡 ⁢ 𝑏 ⁢ 𝑡 − 1 ∈ 𝐿 ⁢ 𝑁 , 𝑡 2 ⁢ 𝑏 ⁢ 𝑡 − 2 ∈ 𝐿 ⁢ 𝑁 , and  𝑑 ⁢ 𝑡 ⁢ 𝑏 ⁢ 𝑡 − 1 ⁢ 𝑑 − 1 ∈ 𝐿 ⁢ 𝑁 hold for any 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝔼 ) .

Thus, 𝐻 ¯

𝐻 and  𝐾 ¯

𝐾 . When 𝔎 satisfies a non-trivial identity, the proof is similar. ∎

6.Proof of Theorems 4–6 and Corollary 2

If 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ , 𝑄  is a normal ( 𝐻 , 𝐾 , 𝜑 ) -compatible subgroup of  𝐵 , and  𝒞 is a class of groups, then we say that 𝑄  is

a)   𝒞 -admissible if there exists a homomorphism of the group

𝔼 𝑄

⟨ 𝐵 / 𝑄 , 𝑡 ; 𝑡 − 1 ( 𝐻 𝑄 / 𝑄 ) 𝑡

𝐾 𝑄 / 𝑄 , 𝜑 𝑄 ⟩

onto a group from  𝒞 acting injectively on  𝐵 / 𝑄 ;

b)  pre- 𝒞 -admissible if 𝐵 / 𝑄 ∈ 𝒞 and  𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄

𝐿 ⁢ 𝑄 / 𝑄 .

The next proposition follows from Theorems 1 and 3 of [44].

Proposition 6.1.

Let 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ , and let 𝒞 be a root class of groups. Suppose also that 𝐻 and  𝐾 are  𝒞 -separable in  𝐵 and each subgroup of  𝒞 ∗ ⁢ ( 𝐵 ) contains a  𝒞 -admissible subgroup. Then 𝔼 is residually a  𝒞 -group if and only if 𝐵 has the same property.

Proposition 6.2.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups closed under taking quotient groups. Suppose also that 𝔉 ∈ 𝒞 and the following conditions hold:

(†)

at least one of the subgroups  ℌ and  𝔎 is normal in  𝔘 or  𝔘 ∈ 𝒞 ;

(‡)

at least one of the subgroups  𝔏 and  𝔉 is normal in  𝔙 or  𝔙 ∈ 𝒞 .

If 𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups and each subgroup of  𝒞 ∗ ⁢ ( 𝐵 ) contains a pre- 𝒞 -admissible subgroup, then 𝔼 and  𝐵 are residually 𝒞 -groups simultaneously.

Proof.

In view of Propositions 3.1 and 6.1, it suffices to show that every pre- 𝒞 -admissible subgroup  𝑄 is  𝒞 -admissible.

Indeed, let the groups  𝔼 𝑄 , ℌ 𝑄 , 𝔎 𝑄 , 𝔏 𝑄 , 𝔉 𝑄 , 𝔘 𝑄 , and  𝔙 𝑄 be defined as in Proposition 4.4. By the latter, there exists a homomorphism of  𝔘 onto the group  𝔘 𝑄 mapping  ℌ and  𝔎 onto  ℌ 𝑄 and  𝔎 𝑄 , respectively. Therefore, if  ℌ is normal in  𝔘 , then ℌ 𝑄 is normal in  𝔘 𝑄 and, hence, the latter is an extension of  ℌ 𝑄 by the group 𝔘 𝑄 / ℌ 𝑄

𝔎 𝑄 ⁢ ℌ 𝑄 / ℌ 𝑄 ≅ 𝔎 𝑄 / ℌ 𝑄 ∩ 𝔎 𝑄 . Since 𝐵 / 𝑄 ∈ 𝒞 , it follows from Proposition 3.4 that Aut 𝐵 / 𝑄 ⁡ ( 𝐻 ⁢ 𝑄 / 𝑄 ) ∈ 𝒞 and  Aut 𝐵 / 𝑄 ⁡ ( 𝐾 ⁢ 𝑄 / 𝑄 ) ∈ 𝒞 . But  Aut 𝐵 / 𝑄 ⁡ ( 𝐾 ⁢ 𝑄 / 𝑄 ) is isomorphic to  𝔎 𝑄 . Hence, ℌ 𝑄 , 𝔎 𝑄 ∈ 𝒞 and  𝔘 𝑄 ∈ 𝒞 because 𝒞 is closed under taking quotient groups and extensions. If  𝔎 is normal in  𝔘 , then the relation 𝔘 𝑄 ∈ 𝒞 is proved in exactly the same way, while if 𝔘 ∈ 𝒞 , this relation follows from the fact that 𝒞 is closed under taking quotient groups.

The inclusion 𝔙 𝑄 ∈ 𝒞 can be verified in a similar way, the only difference is that the relation 𝔉 𝑄 ∈ 𝒞 is ensured by the condition 𝔉 ∈ 𝒞 . Hence, it follows that 𝑄 is  𝒞 -admissible due to Theorem 1. ∎

Proposition 6.3.

[38, Propositions 5.2, 6.1, and 6.3, Theorem 2.2] If 𝒞 is a root class of groups consisting only of periodic groups, then the following statements hold.

1.  The class of  𝒞 -bounded nilpotent groups is closed under taking subgroups and quotient groups.

2.  If the exponent of a  𝒞 -bounded nilpotent group is finite and is a  𝔓 ⁢ ( 𝒞 ) -number, then this group belongs to  𝒞 .

3.  Every 𝒞 -bounded nilpotent group is  𝒞 -quasi-regular with respect to any of its subgroups.

4.  A subgroup of a  𝒞 -bounded nilpotent group  𝑋 is  𝒞 -separable in this group if and only if it is  𝔓 ⁢ ( 𝒞 ) ′ -isolated in  𝑋 .

Proposition 6.4.

If the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞 is a root class of groups, then the following statements hold.

1.  If 𝔙 ∈ 𝒞 , then, for each subgroup 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐿 ) , there exists a subgroup 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐿 ) lying in  𝑅 and such that 𝑆 ⁢ 𝔳

𝑆 for any automorphism 𝔳 ∈ 𝔙 .

2.  If 𝔘 ∈ 𝒞 , then, for each subgroup 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) , there exists a subgroup 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) lying in  𝑅 and such that 𝑆 ⁢ 𝔲

𝑆 for any automorphism 𝔲 ∈ 𝔘 .

3.  Suppose that 𝒞 consists only of periodic groups and  𝑁 is a subgroup of  𝐵 that is locally cyclic or  𝒞 -bounded nilpotent. Then, for each subgroup 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝑁 ) , there exists a subgroup 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝑁 ) lying in  𝑅 and such that 𝑆 ⁢ 𝔞

𝑆 for any automorphism 𝔞 ∈ Aut ⁡ 𝑁 .

Proof.

1.  Let 𝑆

⋂ 𝔳 ∈ 𝔙 𝑅 ⁢ 𝔳 . By Remak’s theorem (see., e.g., [58, Theorem 4.3.9]) the quotient group  𝐿 / 𝑆 can be embedded to the unrestricted direct product of the groups 𝐿 / 𝑅 ⁢ 𝔳 , 𝔳 ∈ 𝔙 , each of which is isomorphic to the  𝒞 -group  𝐿 / 𝑅 . Therefore, it follows from the condition 𝔙 ∈ 𝒞 and the root class definition that 𝐿 / 𝑆 ∈ 𝒞 . It is also clear that 𝑆 ⁢ 𝔳

𝑆 for any 𝔳 ∈ 𝔙 .

2.  Let 𝑆

⋂ 𝔲 ∈ 𝔘 𝑅 ⁢ 𝔲 . As in the proof of Statement 1, it follows from the inclusions 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) and  𝔘 ∈ 𝒞 that 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) . The equality 𝑆 ⁢ 𝔲

𝑆 is obvious for any 𝔲 ∈ 𝔘 .

3.  By Proposition 3.8, the exponent  𝑞 of the  𝒞 -group  𝑁 / 𝑅 is finite. Consider the subgroup 𝑆

sgp ⁡ { 𝑥 𝑞 ∣ 𝑥 ∈ 𝑁 } . Clearly, 𝑆 ⩽ 𝑅 and  𝑆 ⁢ 𝔞

𝑆 for any automorphism 𝔞 ∈ Aut ⁡ 𝑁 . It is also obvious that 𝑞 is a  𝔓 ⁢ ( 𝒞 ) -number and is equal to the exponent of  𝑁 / 𝑆 . Therefore, if  𝑁 is a locally cyclic group, then 𝑁 / 𝑆 is a finite cyclic group, which belongs to  𝒞 by Proposition 3.8. When 𝑁 is a  𝒞 -bounded nilpotent group, 𝑁 / 𝑆  is also a  𝒞 -bounded nilpotent group and  𝑁 / 𝑆 ∈ 𝒞 due to Statements 1 and 2 of Proposition 6.3. ∎

Proposition 6.5.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞  is a root class of groups closed under taking quotient groups, 𝐻 / 𝐿 ∈ 𝒞 , and  𝔉 ∈ 𝒞 . Suppose also that at least one of the following statements holds:

( 𝑎 )

the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 , (†) holds, and  𝔙 ∈ 𝒞 ;

( 𝑏 )

the class  𝒞 consists only of periodic groups, 𝐻  and  𝐾 are locally cyclic subgroups, and the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 ;

( 𝑐 )

the class  𝒞 consists only of periodic groups, (†) and (‡) hold, and  𝐵 is a  𝒞 -bounded nilpotent group.

If 𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups, then 𝔼 and  𝐵 are residually 𝒞 -groups simultaneously.

Proof.

First of all, let us note that, by Proposition 6.3, a  𝒞 -bounded nilpotent group is  𝒞 -quasi-regular with respect to any of its subgroups. It is also known that the automorphism group of a locally cyclic group is abelian (see, e.g., [59, § 113, Exercise 4]). Therefore, (†), (‡), and the  𝒞 -quasi-regularity of  𝐵 with respect to  𝐻 ⁢ 𝐾 hold under any of Statements ( 𝑎 )–( 𝑐 ). This fact and Proposition 6.2 imply that, to end the proof, it suffices to fix a subgroup 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) and show that it contains a pre- 𝒞 -admissible subgroup.

Let 𝑅

𝑀 ∩ 𝐿 . Then 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐿 ) by Proposition 3.2. If  𝐻 and  𝐾 are locally cyclic groups, then 𝐿 is also locally cyclic. By Proposition 6.3, if  𝐵 is a  𝒞 -bounded nilpotent group, then 𝐿 has the same property. Therefore, it follows from Statements 1 and 3 of Proposition 6.4 that there exists a subgroup 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐿 ) lying in  𝑅 and satisfying the equality 𝑆 ⁢ 𝔳

𝑆 for any automorphism 𝔳 ∈ 𝔙 . Since 𝔙

Aut 𝔼 ⁡ ( 𝐿 ) , the subgroup  𝑆 turns out to be normal in  𝔼 and, in particular, is  𝜑 -invariant. The quotient group  𝐻 ⁢ 𝐾 / 𝑆 is an extension of the  𝒞 -group  𝐿 / 𝑆 by a group isomorphic to  𝐻 ⁢ 𝐾 / 𝐿 . The latter, in turn, is an extension of the  𝒞 -group  𝐻 / 𝐿 by a group isomorphic to  𝐻 ⁢ 𝐾 / 𝐻 . The equalities 𝐻 ⁢ 𝜑

𝐾 and  𝐿 ⁢ 𝜑

𝐿 imply that 𝐾 / 𝐿 ≅ 𝐻 / 𝐿 . Since 𝐻 ⁢ 𝐾 / 𝐻 ≅ 𝐾 / 𝐻 ∩ 𝐾

𝐾 / 𝐿 and the class  𝒞 is closed under taking extensions, it follows that 𝐻 ⁢ 𝐾 / 𝑆 ∈ 𝒞 . The  𝒞 -quasi-regularity of  𝐵 with respect to  𝐻 ⁢ 𝐾 and Proposition 3.7 guarantee the existence of a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) such that 𝑁 ∩ 𝐻 ⁢ 𝐾

𝑆 . Let us show that the subgroup 𝑄

𝑀 ∩ 𝑁 is pre- 𝒞 -admissible and, therefore, is the desired one.

Indeed, it follows from the inclusions 𝑀 , 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) and Proposition 3.3 that 𝑄 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) . The relations

𝑆 ⩽ 𝑅

𝐿 ∩ 𝑀

𝐻 ∩ 𝐾 ∩ 𝑀

and  𝑆 ⁢ 𝜑

𝑆 imply that 𝑄 ∩ 𝐻 ⁢ 𝐾

𝑀 ∩ ( 𝑁 ∩ 𝐻 ⁢ 𝐾 )

𝑆 and, therefore,

( 𝑄 ∩ 𝐻 ) ⁢ 𝜑

( 𝑄 ∩ 𝐻 ∩ 𝐻 ⁢ 𝐾 ) ⁢ 𝜑

𝑆 ⁢ 𝜑

𝑆

𝑄 ∩ 𝐾 ∩ 𝐻 ⁢ 𝐾

𝑄 ∩ 𝐾 .

If  𝑥 ∈ 𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄 and  𝑥

ℎ ⁢ 𝑄

𝑘 ⁢ 𝑄 for some ℎ ∈ 𝐻 and  𝑘 ∈ 𝐾 , then

ℎ − 1 ⁢ 𝑘 ∈ 𝑄 ∩ 𝐻 ⁢ 𝐾

𝑆 ⩽ 𝐻 ∩ 𝐾 .

Hence, ℎ , 𝑘 ∈ 𝐻 ∩ 𝐾

𝐿 and  𝑥 ∈ 𝐿 ⁢ 𝑄 / 𝑄 . Thus, 𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄 ⩽ 𝐿 ⁢ 𝑄 / 𝑄 and, because the opposite inclusion is obvious, the subgroup  𝑄 is pre- 𝒞 -admissible. ∎

Proposition 6.6.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) , 𝒞  is a root class of groups closed under taking quotient groups, 𝔉 ∈ 𝒞 , and there exists a homomorphism  𝜎 of  𝐵 onto a group from  𝒞 acting injectively on  𝐿 . Suppose also that at least one of the following statements holds:

( 𝑎 )

the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 , (‡) holds, and  𝔘 ∈ 𝒞 ;

( 𝑏 )

the class  𝒞 consists only of periodic groups, 𝐻  and  𝐾 are locally cyclic subgroups, and the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 ;

( 𝑐 )

the class  𝒞 consists only of periodic groups, (†) and (‡) hold, and  𝐵 is a  𝒞 -bounded nilpotent group;

( 𝑑 )

the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 , (‡) holds, and the group  𝔘 coincides with one of its subgroups  ℌ and  𝔎 .

If 𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups, then 𝔼 and  𝐵 are residually 𝒞 -groups simultaneously.

Proof.

Replacing, if necessary, 𝐻  with  𝐾 , 𝜑  with  𝜑 − 1 , and  𝑡 with  𝑡 − 1 , we can further assume that, if Statement ( 𝑑 ) holds, then 𝔘

ℌ . As in the proof of Proposition 6.5, (†), (‡), and the  𝒞 -quasi-regularity of  𝐵 with respect to  𝐻 ⁢ 𝐾 hold under any of Statements ( 𝑎 )–( 𝑑 ). Therefore, it suffices to show that each subgroup 𝑀 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) contains some pre- 𝒞 -admissible subgroup.

Let 𝑃

𝑀 ∩ ker ⁡ 𝜎 and  𝑅

( 𝑃 ∩ 𝐻 ) ∩ ( 𝑃 ∩ 𝐾 ) ⁢ 𝜑 − 1 . It follows from the inclusions 𝑀 , ker ⁡ 𝜎 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) and Propositions 3.2 and 3.3 that

𝑃 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) , 𝑃 ∩ 𝐻 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) , 𝑃 ∩ 𝐾 ∈ 𝒞 ∗ ⁢ ( 𝐾 ) , ( 𝑃 ∩ 𝐾 ) ⁢ 𝜑 − 1 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) ,

and  𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) . Let us show that there exists a subgroup 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) lying in  𝑅 and satisfying the equality 𝑆 ⁢ 𝔲

𝑆 for any automorphism 𝔲 ∈ 𝔘 .

If 𝐵 is a  𝒞 -bounded nilpotent group, then 𝐻 has the same property by Proposition 6.3. Therefore, when one of Statements ( 𝑎 )–( 𝑐 ) holds, the desired subgroup  𝑆 exists due to Statements 2 and 3 of Proposition 6.4. Let Statement ( 𝑑 ) hold. Since the class  𝒞 is closed under taking quotient groups, it follows from the relations

𝐻 ⁢ 𝐾 / 𝑅 ⁢ 𝐾 ≅ 𝐻 / 𝑅 ⁢ ( 𝐻 ∩ 𝐾 ) ≅ ( 𝐻 / 𝑅 ) / ( 𝑅 ⁢ 𝐿 / 𝑅 )

that 𝑅 ⁢ 𝐾 ∈ 𝒞 ∗ ⁢ ( 𝐻 ⁢ 𝐾 ) . The  𝒞 -quasi-regularity of  𝐵 with respect to  𝐻 ⁢ 𝐾 guarantees the existence of a subgroup 𝑇 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) such that 𝑇 ∩ 𝐻 ⁢ 𝐾 ⩽ 𝑅 ⁢ 𝐾 . Let 𝑆

𝑃 ∩ 𝑇 ∩ 𝐻 . Then 𝑆 ⩽ 𝑅 ⁢ 𝐾 and  𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐻 ) due to Propositions 3.2 and 3.3. If  𝑠 ∈ 𝑆 , then 𝑠

𝑟 ⁢ 𝑘 for suitable 𝑟 ∈ 𝑅 and  𝑘 ∈ 𝐾 , and it follows from the inclusions 𝑅 , 𝑆 ⩽ 𝑃 ∩ 𝐻 that 𝑘 ∈ 𝑃 ∩ 𝐻 ∩ 𝐾 ⩽ ker ⁡ 𝜎 ∩ 𝐿

1 . Hence, 𝑆 ⩽ 𝑅 . Since 𝑃 , 𝑇 , and  𝐻 are normal in  𝐵 , the subgroup  𝑆 has the same property. Therefore, 𝑆 ⁢ 𝔥

𝑆 for each 𝔥 ∈ ℌ , and  𝑆 ⁢ 𝔲

𝑆 for each 𝔲 ∈ 𝔘 because 𝔘

ℌ .

Thus, a subgroup  𝑆 with the required properties always exists. Since Aut 𝐵 ⁡ ( 𝐻 )

ℌ ⩽ 𝔘 and  𝜑 ⁢ Aut 𝐵 ⁡ ( 𝐾 ) ⁢ 𝜑 − 1

𝔎 ⩽ 𝔘 , the equalities 𝑆 ⁢ 𝔥

𝑆 , 𝑆 ⁢ 𝜑 ⁢ 𝔨 ⁢ 𝜑 − 1

𝑆 , and  ( 𝑆 ⁢ 𝜑 ) ⁢ 𝔨

𝑆 ⁢ 𝜑 hold for all 𝔥 ∈ Aut 𝐵 ⁡ ( 𝐻 ) and  𝔨 ∈ Aut 𝐵 ⁡ ( 𝐾 ) . Hence, 𝑆  and  𝑆 ⁢ 𝜑 are normal in  𝐵 . It follows from the relations

𝑆 ⩽ 𝑅 ⩽ 𝑃 ∩ 𝐻 , 𝑆 ⁢ 𝜑 ⩽ 𝑅 ⁢ 𝜑 ⩽ 𝑃 ∩ 𝐾 , 𝑃 ∩ 𝐻 ∩ 𝐾 ⩽ ker ⁡ 𝜎 ∩ 𝐿

1

that 𝑆 ∩ 𝐾

1

𝑆 ⁢ 𝜑 ∩ 𝐻 . Therefore, 𝑆 ⋅ 𝑆 ⁢ 𝜑 ∩ 𝐻

𝑆 and  𝑆 ⋅ 𝑆 ⁢ 𝜑 ∩ 𝐾

𝑆 ⁢ 𝜑 .

The group 𝐻 ⁢ 𝐾 / 𝑆 ⋅ 𝑆 ⁢ 𝜑 is an extension of  𝑆 ⁢ 𝐾 / 𝑆 ⋅ 𝑆 ⁢ 𝜑 by a group isomorphic to  𝐻 ⁢ 𝐾 / 𝑆 ⁢ 𝐾 , and the class  𝒞 is closed under taking extensions and quotient groups. Therefore, it follows from the relations

𝑆 ⁢ 𝐾 / 𝑆 ⋅ 𝑆 ⁢ 𝜑 ≅ 𝐾 / 𝑆 ⁢ 𝜑 ⁢ ( 𝐾 ∩ 𝑆 ) ≅ ( 𝐾 / 𝑆 ⁢ 𝜑 ) / ( 𝑆 ⁢ 𝜑 ⁢ ( 𝐾 ∩ 𝑆 ) / 𝑆 ⁢ 𝜑 ) ,

𝐻 ⁢ 𝐾 / 𝑆 ⁢ 𝐾 ≅ 𝐻 / 𝑆 ⁢ ( 𝐻 ∩ 𝐾 ) ≅ ( 𝐻 / 𝑆 ) / ( 𝑆 ⁢ 𝐿 / 𝑆 ) ,

and 𝐾 / 𝑆 ⁢ 𝜑 ≅ 𝐻 / 𝑆 ∈ 𝒞 that 𝐻 ⁢ 𝐾 / 𝑆 ⋅ 𝑆 ⁢ 𝜑 ∈ 𝒞 . Since 𝑆 ⋅ 𝑆 ⁢ 𝜑 is normal in  𝐵 and the latter is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 , Proposition 3.7 implies the existence of a subgroup 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) such that 𝑁 ∩ 𝐻 ⁢ 𝐾

𝑆 ⋅ 𝑆 ⁢ 𝜑 . Let 𝑄

𝑃 ∩ 𝑁 . Then 𝑄 ⩽ 𝑃 ⩽ 𝑀 , and it follows from Proposition 3.3 and the inclusions 𝑃 , 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) that 𝑄 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) . Let us show that 𝑄 is pre- 𝒞 -admissible.

The relations 𝑆 ⩽ 𝑃 ∩ 𝐻 and  𝑆 ⁢ 𝜑 ⩽ 𝑃 ∩ 𝐾 imply that

𝑄 ∩ 𝐻 ⁢ 𝐾

𝑃 ∩ ( 𝑁 ∩ 𝐻 ⁢ 𝐾 )

𝑆 ⋅ 𝑆 ⁢ 𝜑 .

Since 𝑆 ⋅ 𝑆 ⁢ 𝜑 ∩ 𝐻

𝑆 and  𝑆 ⋅ 𝑆 ⁢ 𝜑 ∩ 𝐾

𝑆 ⁢ 𝜑 , as proven above, the equalities

( 𝑄 ∩ 𝐻 ) ⁢ 𝜑

( 𝑄 ∩ 𝐻 ∩ 𝐻 ⁢ 𝐾 ) ⁢ 𝜑

𝑆 ⁢ 𝜑

𝑄 ∩ 𝐾 ∩ 𝐻 ⁢ 𝐾

𝑄 ∩ 𝐾

hold. If  𝑥 ∈ 𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄 and  𝑥

ℎ ⁢ 𝑄

𝑘 ⁢ 𝑄 for some ℎ ∈ 𝐻 and  𝑘 ∈ 𝐾 , then

ℎ − 1 ⁢ 𝑘 ∈ 𝑄 ∩ 𝐻 ⁢ 𝐾

𝑆 ⋅ 𝑆 ⁢ 𝜑

and  ℎ − 1 ⁢ 𝑘

𝑠 ⁢ 𝑠 ′ for suitable 𝑠 ∈ 𝑆 and  𝑠 ′ ∈ 𝑆 ⁢ 𝜑 . Therefore,

ℎ ⁢ 𝑠

𝑘 ⁢ ( 𝑠 ′ ) − 1 ∈ 𝐻 ∩ 𝐾

𝐿 ,

ℎ ∈ 𝐿 ⁢ 𝑆 ⩽ 𝐿 ⁢ 𝑄 , and  𝑥 ∈ 𝐿 ⁢ 𝑄 / 𝑄 . Thus, 𝐻 ⁢ 𝑄 / 𝑄 ∩ 𝐾 ⁢ 𝑄 / 𝑄

𝐿 ⁢ 𝑄 / 𝑄 and, hence, 𝑄 is pre- 𝒞 -admissible. ∎

Obviously, if  𝒞 is a root class of groups, then the inclusion 𝔉 ∈ 𝒞 is guaranteed by the condition 𝔙 ∈ 𝒞 . Therefore, Theorem 4 is a special case of Proposition 6.7 below, which, in turn, follows from Propositions 6.5 and 6.6. To anticipate possible questions from the reader, we note that Propositions 6.7 and 6.8 use Statements  ( 𝛼 ) and  ( 𝛽 ) from Theorem 4.

Proposition 6.7.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups closed under taking quotient groups. Suppose also that 𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 and at least one of the following statements holds:

( 𝑎 )

𝔙 ∈ 𝒞 and  ( 𝛼 ) and (†) hold;

( 𝑏 )

𝔘 , 𝔉 ∈ 𝒞 and  ( 𝛽 ) and (‡) hold.

If 𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups, then  𝔼 and  𝐵 are residually 𝒞 -groups simultaneously.

Theorem 5 follows from Propositions 3.2, 6.5, 6.6 and Theorem 3.

Proof of Theorem 6.

1.  Since 𝐵 / 𝐻 and  𝐵 / 𝐾 are residually 𝒞 -groups, the subgroups  𝐻 and  𝐾 are  𝒞 -separable in  𝐵 by Proposition 3.1. It easily follows that 𝐿 is also 𝒞 -separable in  𝐵 and, again by Proposition 3.1, 𝐵 / 𝐿  is residually a  𝒞 -group. Hence, if the subgroup  𝐻 / 𝐿 is finite, then it belongs to  𝒞 due to Proposition 3.3. Conversely, if the locally cyclic group  𝐻 / 𝐿 belongs to  𝒞 , then it has a finite exponent by Proposition 3.8 and, therefore, is finite.

2.  If ( 𝛽 ) holds, then the locally cyclic group  𝐿 can be embedded in a  𝒞 -group. As above, this implies its finiteness. The opposite statement follows from Proposition 3.3.

3.  Necessity is ensured by Proposition 3.2 and Theorem 3. To prove sufficiency, let us show that 𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 . Then the residual 𝒞 -ness of  𝔼 will follow from Statement 1 of this theorem and Proposition 6.5.

As noted in the proof of the latter, the quotient group  𝐻 ⁢ 𝐾 / 𝐿 is an extension of  𝐻 / 𝐿 by a group isomorphic to  𝐻 / 𝐿 and, therefore, is finite. By the arguments used to verify Statement 1, the residual 𝒞 -ness of  𝐵 / 𝐻 and  𝐵 / 𝐾 implies the residual 𝒞 -ness of  𝐵 / 𝐿 . Due to Proposition 3.3, it follows that there exists a subgroup 𝑆 / 𝐿 ∈ 𝒞 ∗ ⁢ ( 𝐵 / 𝐿 ) satisfying the condition 𝑆 / 𝐿 ∩ 𝐻 ⁢ 𝐾 / 𝐿

1 . Clearly, 𝑆 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) and  𝑆 ∩ 𝐻 ⁢ 𝐾 ⩽ 𝐿 .

Now, if  𝑀 ∈ 𝒞 ∗ ⁢ ( 𝐻 ⁢ 𝐾 ) and  𝑄

𝑀 ∩ 𝐿 , then 𝑄 ∈ 𝒞 ∗ ⁢ ( 𝐿 ) by Proposition 3.2. Since 𝐵 is  𝒞 -quasi-regular with respect to  𝐿 , there exists a subgroup 𝑅 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) such that 𝑅 ∩ 𝐿 ⩽ 𝑄 . Let 𝑁

𝑅 ∩ 𝑆 . Then 𝑁 ∈ 𝒞 ∗ ⁢ ( 𝐵 ) by Proposition 3.3 and

𝑁 ∩ 𝐻 ⁢ 𝐾

𝑅 ∩ 𝑆 ∩ 𝐻 ⁢ 𝐾 ⩽ 𝑅 ∩ 𝐿 ⩽ 𝑄 ⩽ 𝑀 .

Thus, the group  𝐵 is  𝒞 -quasi-regular with respect to  𝐻 ⁢ 𝐾 , as required.

4.  Sufficiency follows from Proposition 3.3, which ensures that ( 𝛽 ) holds, and Proposition 6.6. Let us prove necessity.

By Proposition 3.3, since 𝔼 is residually a  𝒞 -group, it has a homomorphism onto a group from  𝒞 acting injectively on the finite subgroup  𝐿 . This fact and Proposition 3.4 imply that 𝔙

Aut 𝔼 ⁡ ( 𝐿 ) ∈ 𝒞 and  𝔉 ∈ 𝒞 . As above, the residual 𝒞 -ness of the groups  𝐵 , 𝐵 / 𝐻 , and  𝐵 / 𝐾 is ensured by Proposition 3.2 and Theorem 3. ∎

Corollary 2 can be deduced either from Theorems 3–6 and Propositions 3.1, 3.2, and 6.3, or from Proposition 6.8 below. The second method uses the fact that the automorphism group of a locally cyclic group is abelian, which is already mentioned in the proof of Proposition 6.5.

Proposition 6.8.

Suppose that the group 𝔼

⟨ 𝐵 , 𝑡 ; 𝑡 − 1 𝐻 𝑡

𝐾 , 𝜑 ⟩ satisfies  ( ∗ ) and  𝒞  is a root class of groups consisting only of periodic groups and closed under taking quotient groups. Suppose also that 𝔉 ∈ 𝒞 and  𝐵 is a  𝒞 -bounded nilpotent group. Finally, let (†), (‡), and at least one of Statements  ( 𝛼 ) and  ( 𝛽 ) hold. Then 𝔼 is residually a  𝒞 -group if and only if the subgroups  { 1 } , 𝐻 , and  𝐾 are  𝔓 ⁢ ( 𝒞 ) ′ -isolated in  𝐵 .

Proof.

First of all, let us note that, by Proposition 6.3, the subgroups  { 1 } , 𝐻 , and  𝐾 are  𝔓 ⁢ ( 𝒞 ) ′ -isolated in  𝐵 if and only if they are  𝒞 -separable in this group. Due to Proposition 3.1, the latter property is equivalent to the residual 𝒞 -ness of the groups  𝐵 , 𝐵 / 𝐻 , and  𝐵 / 𝐾 . Therefore, necessity follows from Proposition 3.2 and Theorem 3, while sufficiency can be deduced from Propositions 6.5 and 6.6. ∎

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