Group varieties not closed under cellular covers and localizations

Theorem 1.1.

It is clear that the only abelian subvarieties of Bp are the trivial variety {1} and the variety Ap of abelian groups of exponent p. But the variety

contains (for any prime p>1075) all abelian groups. Thus Bp is not cellular closed.

Question 1.

Are there 2ℵ0 varieties of groups which are not closed under cellular covers?

Definition 2.1.

A homomorphism e:H→G is a cellular cover of G if every homomorphism φ:H→G factors uniquely through e, i.e., there is a unique homomorphism φ¯:H→H such that φ¯⁢e=φ:

Lemma 2.2.

If the group B satisfies conditions (i) and (ii), then B is simple.

Proof.

To show that B is simple, suppose that N≠B is a nontrivial normal subgroup of B. It follows that N is a proper subgroup of order p. Let g be any element in B∖N. Because N is normal and g is of finite order, 〈g,N〉=〈g〉⁢N is a proper finite subgroup of order >p, a contradiction. ∎

Theorem 2.3.

If Bp=F/N is a free presentation of Bp, then N/[F,N] is a free abelian group of countable rank.

Remark 2.4.

Let X={x1,x2} be a set of two generators and F=F⁢(X) the associated free group. We have an aspherical representation ℬp=〈X|ℛ〉 for a suitable set ℛ of relators. In particular, we have ℬp=F/N with N=ℛF the normal closure of the set of relators ℛ. It is clear that N/[F,N] is abelian, however, the asphericity of the presentation implies that {r=1∣r∈ℛ} is an independent set of relations and, consequently, that {r⁢[F,N]∣r∈ℛ} is a basis of N/[F,N].

Lemma 2.5.

The group Bp=F/N has a central extension

where K:=(F′∩N)/[F,N], A:=F′/[F,N], and

is a perfect cover.

Remark 2.6.

This central extension is also known as the universal perfect cover of ℬp and has a number of important properties; see for example [2] and [17]. In a similar fashion, a universal perfect cover is possible for any perfect group. Observe here that ℬp is clearly perfect as it is a non-abelian simple group.

Lemma 2.7.

The group Bp has a cellular cover e:A→Bp, where Bp=F/N, A=F′/[F,N], and

The group A contains a copy of Z as subgroup.

Proof.

We have to show that e:A→ℬp induces a bijection

First, we claim that the map e* is surjective: Let φ∈Hom⁡(A,ℬp). We must find φ¯∈Hom⁡(A,A) such that φ¯⁢e=φ. If φ=0, then we choose φ¯=0. So φ¯⁢e=φ clearly holds. Next, we assume that φ≠0, then 1≠A⁢φ⊆ℬp. If A⁢φ≠ℬp, then, by the choice of ℬp, A⁢φ is cyclic hence abelian. Since A is perfect, we have that

i.e., A⁢φ is perfect as well. Consequently, we obtain that A⁢φ=[A⁢φ,A⁢φ]=1 which contradicts the assumption that φ≠0. Thus, we now have that A⁢φ=ℬp, i.e., φ is surjective. As ℬp is simple, we know that 𝔷⁢ℬp=1. Moreover, we can see that (𝔷⁢A)⁢φ⊆𝔷⁢ℬp=1 because φ is surjective. From the above central extension, we obtain that K⊆𝔷⁢A⊆Ker⁡φ and φ=e⁢β follows for some β∈Hom⁡(ℬp,ℬp). Next, we claim that β∈Aut⁡(ℬp). Since φ≠0 and ℬp is simple, we have that Ker⁡β=1, which shows that β is injective. We can see that β is also surjective by using the fact that φ is surjective. Thus, we finally have that β∈Aut⁡(ℬp), hence Ker⁡φ=Ker⁡e=K.

We next consider the following diagram:

where the map λ is an epimorphism defined by sending f⁢[F,N]∈F/[F,N] to f⁢N∈F/N. By the help of [17, Lemma 2.4.1], we obtain a homomorphism α:F/[F,N]→A such that the above diagram commutes, i.e., α⁢e=λ⁢β. Recall that A=F′/[F,N]⊆F/[F,N]. Hence, α:F/[F,N]→A can be restricted to φ¯:=α↾A. Then φ¯∈Hom⁡(A,A). But e=λ↾A, and hence φ=e⁢β=φ¯⁢e.

Next, we will show that e* is injective: Let φ1,φ2∈Hom⁡(A,A) such that φ1⁢e=φ2⁢e. Define

It is easy to see that A⁢ψ⊆Ker⁡e=K. For a,b∈A, we obtain that

Using the fact that K=𝔷⁢A, since b⁢ψ∈K and a-1⁢φ2∈A, we then obtain

i.e., ψ∈Hom⁡(A,K). Moreover, as A is perfect and A⁢ψ⊆K=𝔷⁢A is abelian, we obtain A⁢ψ=[A,A]⁢ψ=[A⁢ψ,A⁢ψ]=1. As a result, ψ=0 and then

for all a∈A. This shows that φ1=φ2. Thus e* is injective, as desired.

Therefore, we have that e:A→ℬp is a cellular cover of ℬp. Note that A contains K=H2⁢(ℬp,ℤ) which is free abelian of infinite countable rank. ∎

Remark 2.8.

In fact, Lemma 2.5 provides the universal central extension of the perfect group ℬp. In this context, Lemma 2.7 gives a more direct proof of the general result [8, Lemma 3.10] that every universal central extension corresponds to a surjective cellular cover.

Theorem 2.9.

There exist countably many pairwise distinct varieties of groups which are neither cellular closed nor finitely based.

Lemma 3.1.

Let e be a positive integer, and assume that there exists an infinite series of finite groups Ti,i∈ω, such that for each i,

1. Ti is from a fixed locally finite variety 𝔙⊆𝔅e,

2. Ti is monolithic, and

3. Ti is not isomorphic to any factor of Tj for i≠j.

If p is a prime not dividing e, then the product variety Ap⁢V has 2ℵ0 pairwise distinct subvarieties.

Remark 3.2.

Note that 𝔄p⁢𝔙 as product of locally finite varieties is again locally finite, cf. [27, Theorem 14.3.1].

Corollary 3.3.

The following statements hold.

1. For any e=8⁢p1, where p1 is an odd prime, the conditions of Lemma 3.1 are satisfied by a suitable series Ti, i∈ω, of finite solvable groups of length 4.

2. For any distinct odd primes p1 and p2 there exist 2ℵ0 pairwise distinct subvarieties of the locally finite variety of length 5 solvable groups of exponent 8⁢p1⁢p2.

Theorem 3.4.

There exist 2ℵ0 pairwise distinct varieties of groups which are not closed under cellular covers.

Proof.

Let p3>1075 be a prime, and let 𝔅p3 be the associated Burnside variety and ℬp3∈𝔅p3 the Burnside group mentioned in Section 2. Next, we apply the idea of enlarging two given varieties 𝔘 and 𝔙 by considering the product variety 𝔘⁢𝔙, which consists of all extensions of a group in 𝔘 by a group in 𝔙.

Since 𝔅p3 is not the variety of all groups, by applying the right cancellation law for product varieties, we have that 𝔙⁢𝔅p3≠𝔙′⁢𝔅p3 for any varieties 𝔙≠𝔙′. See [21] for more details on varieties of groups.

Thus, there remain 2ℵ0 pairwise distinct product varieties 𝔙⁢𝔅p3, where 𝔙 is a variety obtained from Corollary 3.3 (b). It is clear that 𝔙⁢𝔅p3 contains the group ℬp3, as ℬp3∈𝔅p3⊆𝔙⁢𝔅p3. Since 𝔙 is of exponent 8⁢p1⁢p2 and 𝔅p3 is of exponent p3, the product variety 𝔙⁢𝔅p3 is of exponent 8⁢p1⁢p2⁢p3. But, as shown in Lemma 2.7, there exists a cellular cover e:A→ℬp3 such that ℤ⊆A. Therefore, A cannot be of finite exponent and obviously is not in the product variety 𝔙⁢𝔅p3. This shows the existence of 2ℵ0 pairwise distinct varieties which are not closed under cellular covers. ∎

Corollary 3.5.

There exist 2ℵ0 pairwise distinct varieties of groups which are neither cellular closed nor finitely based.

Proof.

From Theorem 3.4 we obtain 2ℵ0 varieties which are not cellular closed. As there are only countably many finitely based varieties, there actually must exist 2ℵ0 pairwise distinct varieties which are neither cellular closed nor finitely based. ∎

Remark 3.6.

Observe that the existence of 2ℵ0 pairwise distinct not finitely based varieties in Corollary 3.5 is derived only implicitly by set-theoretic considerations, while in Theorem 2.9 countably many such varieties are explicitly given with the help of a suitable system of group laws (2.2).

Definition 4.1.

A group homomorphism e:G→H is a localization of G if every homomorphism φ:G→H factors uniquely through e, i.e., there is a unique homomorphism φ¯:H→H such that e⁢φ¯=φ. It is equivalent to saying that e:G→H is a localization of G if the following diagram commutes:

In a slight abuse of notation, we will often identify e:G→H with the group H and therefore call H a localization of G.

Definition 4.2.

We say that a group G is suitable if the following conditions hold:

1. G≠1 is finite with trivial center.

2. Aut⁡(G) is complete.

3. If G¯⊆Aut⁡(G) and G¯≅G, then G¯=Inn(G).

Theorem 4.3.

Let A be a family of suitable groups and μ an infinite cardinal such that μℵ0=μ. Then we can find a group H of cardinality λ=μ+ such that the following holds:

1. The group H is simple. Moreover, if 1≠h∈H, then any element of H is a product of at most four conjugates of h.

2. Any G∈𝒜 is a subgroup of H and any two different groups in 𝒜 have trivial intersection 1 when considered as subgroups of H. If 𝒜 is not empty, then H⁢[𝒜]=H, where the A-socleH⁢[𝒜] is the subgroup of H generated by all copies of groups G∈𝒜 in H.

3. Any monomorphism φ:G→H, G∈𝒜, is induced by some h∈H, i.e., there is an element h∈H such that φ=h*↾G.

4. If G¯⊆H is an isomorphic copy of some G∈𝒜, then the centralizer CH⁢(G¯) is trivial.

5. Any monomorphism φ:H→H is an inner automorphism, i.e., φ=h* for some h∈H.

6. The group H contains a free subgroup of rank μ.

Lemma 4.4.

Any non-abelian finite simple group is suitable.

Proposition 4.5.

The following statements hold.

1. For any even n≥10 there exists a localization of An with a free subgroup of countable rank.

2. If G is a non-abelian finite simple group and κ an infinite cardinal, then there exists a localization of G with a free subgroup of rank κ.

Proof.

(a) This is an immediate consequence of (b) as An is simple for all n≥5. Alternatively, we may use Libman’s localization ρ:An→SO⁢(n-1,ℝ): We know that SO⁢(3,ℝ), and hence any SO⁢(n,ℝ) for n≥3, contains a free subgroup of rank 2; cf. [16, pp. 469–472]. But it is well known that a free group of rank 2 has a free subgroup of countable rank; cf. [18, Vol. II].

(b) All non-abelian finite simple groups are suitable, by Lemma 4.4. Thus, applying Theorem 4.3 for the choice 𝒜={G} and μ=κℵ0, there exists a group H containing G with the following properties:

1. The group H is simple.

2. Any monomorphism from G to H is a restriction of an inner automorphism of H.

3. The centralizer of G in H is trivial.

4. Any monomorphism from H to H is an inner automorphism of H.

5. The group H contains a free subgroup of rank κℵ0≥κ.

It remains to show that the inclusion e:G→H is a localization of G, i.e., we will show that for any φ:G→H, there exists a unique φ¯:H→H such that e⁢φ¯=φ.

The proof taken from [13, Section 1] is the following. If φ=0, then choose φ¯=0. We obtain G⁢e⁢φ¯=G⁢φ¯=1=G⁢φ. Since H is simple and 1≠G⊆Ker⁡φ¯, the zero map is the only possibility for φ¯.

If φ≠0, then we obtain that Ker⁡φ=1 because Ker⁡φ⊴G and G is simple. Thus, φ is a monomorphism. By (iii), φ is a restriction of an inner automorphism of H, say φ=h¯*↾G, where h¯∈H. Choose φ¯=h¯*. Then e⁢φ¯=φ follows.

Next, assume that φ′:H→H is another homomorphism such that e⁢φ′=φ. Consequently, φ′≠0, and hence φ′ is a monomorphism of H, by (i). Thus, by (v), there exists h∈H such that φ′=h*. We then obtain that

for all g∈G. Hence, h*=h¯* on G, and h-1⁢g⁢h=h¯-1⁢g⁢h¯ for all g∈G. This implies that h¯⁢h-1∈CH⁢(G). Therefore, (iv) implies h¯=h, hence φ¯=φ′. This shows the uniqueness of φ¯ and completes the proof. ∎

Corollary 4.6.

Any non-abelian finite simple group has a localization which contains a free subgroup of arbitrarily large cardinality.

Theorem 4.7.

There exist 2ℵ0 pairwise distinct varieties of groups which are neither closed under localizations nor finitely based.

Proof.

Consider the result by Adjan [1] which shows that, for all odd n≥4381, the set of relations

where r is a prime, is irreducible. In particular, any infinite set of such relations automatically generates a variety which is not finitely based. Let 𝔙 be any such variety. It is easy to see that every group of exponent n satisfies all the laws (4.1), more precisely, 𝔅n⊆𝔙.

Let now n≥4381 be odd and divisible by 15 (e.g., n=4395). Note that A5 is a non-abelian finite simple group with exponent 30. For any element x∈A5, it follows that (xr⁢n)2=x2⁢r⁢n=1 holds because 30∣2n. Thus,

which implies that the relation

is satisfied in A5. This shows that A5 is a member of 𝔙. But, by Corollary 4.6, this gives us that 𝔙 is not closed under localizations as otherwise 𝔙 would contain free groups of arbitrary rank, hence all groups, which is a contradiction. This establishes 2ℵ0 pairwise distinct varieties of groups neither closed under localizations nor finitely based. ∎

Remark 4.8.

It is not apparent whether or not Adjan’s varieties 𝔙 are closed under cellular covers. In particular, our argumentation from Theorem 2.9 fails as 𝔄⊆𝔙 for the variety 𝔄 of all abelian groups, a problem which seems difficult to bypass.

Theorem 5.1.

There exist 2ℵ0 pairwise distinct locally finite varieties of groups which are not closed under localizations.

Proof.

Let S be any non-abelian finite simple group and set 𝔚:=var⁢{S}. Then 𝔚 is locally finite, cf. [21, Theorem 15.71]. Applying the same argument as in Theorem 3.4, we have 𝔙⁢𝔚≠𝔙′⁢𝔚 for any varieties 𝔙≠𝔙′. Thus, there exist 2ℵ0 distinct product varieties 𝔙⁢𝔚, where 𝔙 is a locally finite variety obtained from Corollary 3.3  (b). It is clear that each product variety 𝔙⁢𝔚 contains the group S, as S∈𝔚⊆𝔙⁢𝔚. Furthermore, 𝔙⁢𝔚 is locally finite as it is a product variety of locally finite varieties. The same argument as in Theorem 4.7 applies to show that none of these 2ℵ0 varieties can be closed under localizations. ∎

Remark 5.2.

We make the following remarks.

1. Note that Adjan’s varieties from Theorem 4.7 do contain the variety 𝔄 of all abelian groups and thus are not locally finite.

2. Instead of the product variety 𝔙⁢𝔚, we might use 𝔙∪𝔚, the minimal variety containing both 𝔙 and 𝔚. If p is a prime with gcd⁡(p,|S|)=1 and 𝔙 a variety of exponent pk for some k, then 𝔙∪𝔚 will be the collection of all direct products of a group from 𝔙 and a group from 𝔚. As 𝔙 is uniquely determined by 𝔙∪𝔚 via 𝔙=(𝔙∪𝔚)∩𝔅pk, we have 𝔙∪𝔚≠𝔙′∪𝔚 for any varieties 𝔙≠𝔙′. Thus, the statement would follow from the existence of 2ℵ0 pairwise distinct locally finite varieties of exponent pk.

Corollary 5.3.

There exist 2ℵ0 pairwise distinct locally finite varieties of groups which are neither closed under localizations nor finitely based.

Theorem 5.4.

There exist 2ℵ0 pairwise distinct varieties of groups which are neither closed under cellular covers nor under localizations.

Proof.

Recall from Theorem 3.4 that we obtain 2ℵ0 pairwise distinct product varieties 𝔙⁢𝔅p3 of exponent 8⁢p1⁢p2⁢p3, where 𝔙 is a variety obtained from Corollary 3.3 (b). Let S be any non-abelian finite simple group and set 𝔚:=var⁢{S}. Since S is finite, 𝔚 is a locally finite variety of exponent at most the order of the group S. By multiplying the variety 𝔚 on the right-hand side of each product variety 𝔙⁢𝔅p3, we obtain product varieties of the form 𝔙⁢𝔅p3⁢𝔚 for each variety 𝔙 obtained from Corollary 3.3(b). Because 𝔚 is not the variety of all groups and the 2ℵ0 varieties 𝔙⁢𝔅p3 are all pairwise distinct, the right cancellation law for product varieties can be applied to assure that there still remain 2ℵ0 pairwise distinct product varieties of the form 𝔙⁢𝔅p3⁢𝔚.

However, we see that the two groups ℬp3 and S are members of each variety 𝔙⁢𝔅p3⁢𝔚 as ℬp3∈𝔅p3⊆𝔙⁢𝔅p3⊆𝔙⁢𝔅p3⁢𝔚 and S∈𝔚⊆𝔙⁢𝔅p3⁢𝔚. From Section 2 we know that ℬp3 has cellular covers of infinite exponent and from Section 4 we know that S has localizations of infinite exponent while each variety 𝔙⁢𝔅p3⁢𝔚 is of finite exponent (at most 8⁢p1⁢p2⁢p3⁢q where q is the exponent of 𝔚). It follows that each product variety 𝔙⁢𝔅p3⁢𝔚 is neither closed under taking cellular covers nor closed under taking localizations. ∎