Certain residual properties of HNN-extensions with normal associated subgroups

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.

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.

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 .

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:

1. U = H or U = K ;

2. 𝐻 and 𝐾 satisfy a non-trivial identity;

3. ℌ satisfies a non-trivial identity;

4. 𝔎 satisfies a non-trivial identity.

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:

1. H / L ∈ C ;

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

Suppose that

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

with the coinciding subgroups H = A = K associated by 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,

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

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

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.

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. If 𝐵 is 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 .

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, suppose that at least one of the following statements hold:

1. U , V ∈ C ;

2. U = H or U = K , and V ∈ C ;

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

Proposition 3.1 ([47, 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 ([47, 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 ([47, Proposition 2])

Suppose that 𝒞 is a class of groups closed under taking 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 ([54, 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 .

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. By 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. ∎

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 well-defined surjective homomorphism.

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. ∎

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 [47, 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 ) ) [56, Proposition 8].

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.

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.

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. ∎

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 follow from Proposition 3.3. ∎

Proposition 4.1 ([54, Theorem 1])

If 𝒞 is a root class of groups,

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.

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

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

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 [20] 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 [1].

Let α gen be the mapping of the generators of 𝔓 acting as the isomorphisms β i − 1 ⁢ β i + 1 , i ∈ I . Clearly, α gen 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 gen 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 gen 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

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

The relations

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

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

Let M = M 0 ∩ M − 1 . It follows from Proposition 3.3 and the above that

and M ∈ C ∗ ⁢ ( P ) . Since ⟨ α ⟩ ≅ I ∈ C , we have that 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. ∎

Suppose that the group E = ⟨ B , t ; t − 1 H t = K , φ ⟩ satisfies (∗) and 𝒞 is a root class of groups. If

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

is defined. Consider the following groups:

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,

Since

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

that the subgroup V = Z E ⁢ ( L ) ∩ F is normal in 𝑈,

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 ) , 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 (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

By Proposition 3.6, the relations

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

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

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

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

is defined. It follows from Proposition 3.6 that the map

given by the rule x ̂ | H ↦ x ⁢ ρ Q , Q ̂ | H ⁢ ρ Q , Q , x ∈ B , is a surjective homomorphism. Clearly,

Therefore, the homomorphism ρ Q , Q ̄ is as desired.

(2) The equality L ⁢ φ = L and the definition of φ Q imply that

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 (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 Theorem 2 (2) to prove the residual 𝒞-ness of 𝔼. ∎

Proposition 5.1 ([47, 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 ([22, 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

respectively, and at least one of the following statements holds:

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

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

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 .

By Proposition 3.1, to complete 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 the 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 [34, Lemma 2], it satisfies a non-trivial identity of the form

where n ⩾ 1 , ε 1 , … , ε n = ± 1 , and

Let us assume that there exist elements u 1 , u 2 , v ∈ E with the following properties:

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

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

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

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.