On the exponent of Bogomolov multipliers

Primož Moravec

doi:10.1515/jgth-2018-0041

Article Publicly Available

On the exponent of Bogomolov multipliers

Primož Moravec

Published/Copyright: February 15, 2019

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From the journal Journal of Group Theory Volume 22 Issue 3

Abstract

We prove that if G is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of G in the following four cases: (i) G is metabelian, (ii) exp⁡G=4, (iii) G is nilpotent of class ≤5, or (iv) G is a 4-Engel group.

1 Introduction

Let G be a group given by a free presentation G=F/R. Then the Schur multiplier of G can be defined via Hopf’s formula as M⁡(G)=([F,F]∩R)/[F,R]. It is known that if G is a finite group, then

M⁡(G)≅H2⁡(G,ℤ)≅H2⁡(G,ℚ/ℤ);

see, for example, Beyl and Tappe’s book [3] for details and applications of Schur multipliers.

A significant part of the theory of Schur multipliers consists of estimating their size, rank, or exponent. Focusing on the latter, Schur himself [23] showed that if G is a finite group, then (exp⁡M⁡(G))2 divides |G|. On the other hand, it was shown in [14] that, for every finite group G, the exponent of M⁡(G) can be bounded in terms of exp⁡G only. Good bounds of this kind are however still out of reach. In practice it often happens that exp⁡M⁡(G) divides exp⁡G. One may conjecture that this is always the case, yet a counterexample of exponent 4 with Schur multiplier of exponent 8 was constructed by Bayes, Kautsky, and Wamsley [2]. We mention that there are no known examples of odd order groups G with exp⁡M⁡(G)>exp⁡G, though it seems plausible that they exist. On the other hand, exp⁡M⁡(G) always divides exp⁡G at least when G is of one of the following types: nilpotent of class ≤3 [12, 15], powerful p-group [13], a p-group of maximal class [17]. In addition to that, we have the following.

Theorem 1.1 ([14, 15, 16]).

Let G be a group of finite exponent.

If G is nilpotent of class ≤4, then exp⁡M⁡(G) divides 2⋅exp⁡G.
If G is a 4 -Engel group, then exp⁡M⁡(G) divides 10⋅exp⁡G.
If exp⁡G=4, then exp⁡M⁡(G) divides 8.
If G is metabelian, then exp⁡M⁡(G) divides (exp⁡G)2.

There are numerous other estimates for exp⁡M⁡(G), but we do not go further into this here; see, for example, Sambonet’s paper [22] for a short review of such results.

Given a free presentation G=F/R, define

B0⁡(G)=([F,F]∩R)/(〈K⁡(F)∩R〉),

G⋏G=[F,F]/(〈K⁡(F)∩R〉),

G∧G=[F,F]/[F,R],

where K⁡(F) is the set of commutators in F. It is shown in [18] that if G is a finite group and V a faithful representation of G over ℂ, then the dual of B0⁡(G) is naturally isomorphic to the unramified Brauer group Hnr2⁡(ℂ⁢(V)G,ℚ/ℤ) introduced by Artin and Mumford [1]. This invariant represents an obstruction to Noether’s problem [20] asking as to whether the field ℂ⁢(V)G is purely transcendental over ℂ. The aforementioned result of [18] is based on a result of Bogomolov [4] who showed that Hnr2⁡(ℂ⁢(V)G,ℚ/ℤ) is naturally isomorphic to the intersection of the kernels of restriction maps H2⁡(G,ℚ/ℤ)→H2⁡(A,ℚ/ℤ), where A runs through all (two-generator) abelian subgroups of G. The latter group is also known as the Bogomolov multiplier of G. Here we use the same name for B0⁡(G).

Our main result is the following:

Theorem 1.2.

Let G be a group of finite exponent. If G satisfies one of the following properties, then exp⁡(G⋏G) divides exp⁡G:

nilpotent of class ≤5,
metabelian,
4 -Engel,
exp⁡G=4.

This result thus complements Theorem 1.1. It also implies that, in the cases listed in Theorem 1.2, it always happens that exp⁡B0⁡(G) divides exp⁡G. Note that the question of determining exp⁡B0⁡(G) was recently addressed by García-Rodríguez, Jaikin-Zapirain, and Jezernik [5, Theorem 6].

In view of the above result and further extensive computational evidence, we pose the following conjecture.

Conjecture 1.3.

Let G be a finite group. Then exp⁡B0⁡(G) divides exp⁡G, and thus, in particular, exp⁡M⁡(G) divides (exp⁡G)2.

In order to justify this conjecture, we first mention that Saltman [21] showed that for every p>2, there exists a p-group of exponent p with non-trivial Bogomolov multiplier of exponent p, and thus the bound given in the conjecture is sharp. On the other hand, if exp⁡G is not prime, then B0⁡(G) is usually small (or of small exponent), compared to G. Fernandez-Alcober and Jezernik [10] recently constructed examples of finite p-groups G of maximal class with exp⁡B0⁡(G)≈exp⁡G, and it appears that this might be close to the worst case. In fact, we have not been able to find a group G with exp⁡(G⋏G)>exp⁡G, so it may be possible that even the stronger conjecture with B0⁡(G) replaced by G⋏G might hold true.

The proof of Theorem 1.2 goes roughly as follows: First, one may assume without loss of generality that G is a finite group. By [11], there exists a so-called CP cover H of G (see Section 2 for the details) whose main feature is that G⋏G is isomorphic to [H,H], and all commutator relations of G lift to commutator relations in H. The calculations are then performed in [H,H]. In the metabelian case, the proof is fairly straightforward. In the remaining cases, it relies on a careful examination of the power structure of the lower central series of H, which uses information on the free groups in a given variety. For class ≤5 groups, this can be obtained using elementary commutator calculus. In the exponent 4 and 4-Engel case, we mainly use M. Hall’s description of the free 3-generator group of exponent 4 [7], and Nickel’s computations of free 4-Engel groups of low ranks [19], along with Havas and Vaughan-Lee’s proof of local nilpotency of 4-Engel groups [8].

2 Preliminaries

2.1 CP covers

Let G be a group and Z a G-module. Denote by e=(χ,H,π) the extension

1⟶Z⁢⟶𝜒⁢H⁢⟶𝜋⁢G⟶1

of Z by G. Following [18], we say that e is a CP extension if commuting pairs of elements of G have commuting lifts in H. If e is a central extension, we say that it is a stem extension [3, p. 38] if Zχ≤[H,H]. A stem central CP extension of Z by G, where |Z|=|B0⁡(G)|, is called a CP cover of G. CP covers are analogs of the usual covers in the theory of Schur multipliers. It is proved in [11, Theorem 4.2] that every finite group has a CP cover. It also follows from [11, Proposition 3.3] that if e=(χ,H,π) of the above form is a CP cover of G, then Zχ∩K⁡(H)=1. This in particular implies that any commutator law satisfied by G is also satisfied by H.

2.2 Collection process

We recall [6, Theorem 11.2.4] that Hall’s basis theorem implies that if F is a free nilpotent group of class c and a,b∈F, then the word (a⁢b)n, where n is a non-negative integer, can be written uniquely as a product c1n1⁢c2n2⁢⋯⁢ctnt, where ci are basic commutators in {a,b} of weights 1,2,…,c, and

ni=b1⁢(n1)+b2⁢(n2)+…+br⁢(nr),

where r is the weight of ci and bj are non-negative integers not depending on n. Specifically, we will be interested in the case when F is free nilpotent of class 6. We need to determine the coefficients bi explicitly, and this can be done using the collection process described in [6, Section 12.3]. We omit the details regarding calculations and only record the values of bi for all basic commutators of weight ≤6 in {a,b} in Table 1.

Table 1

Coefficients in exponents of ci.

Commutator ci	b1	b2	b3	b4	b5	b6
a	1
b	1
[b,a]		1
[b,a,a]			1
[b,a,b]		1	2
[b,a,a,a]				1
[b,a,a,b]			2	3
[b,a,b,b]			2	3
[b,a,a,[b,a]]			1	7	6
[b,a,b,[b,a]]			6	18	12
[b,a,a,a,a]					1
[b,a,a,a,b]				3	4
[b,a,a,b,b]			1	6	6
[b,a,b,b,b]				3	4
[b,a,b,[b,a,a]]			4	21	36	20
[b,a,a,a,[b,a]]				3	13	10
[b,a,a,b,[b,a]]			2	24	52	30
[b,a,b,b,[b,a]]			3	27	54	30
[b,a,a,a,a,a]						1
[b,a,a,a,a,b]					4	5
[b,a,a,a,b,b]				3	12	10
[b,a,a,b,b,b]				3	12	10
[b,a,b,b,b,b]					4	5

3 Proof of Theorem 1.2

In what follows, G will be a group of finite exponent satisfying one of the properties listed in Theorem 1.2. In each of those cases, G is then locally finite. As B0 commutes with direct limits [18, Proposition 3.6], one may assume without loss of generality that G is a finite group; furthermore, Bogomolov’s results [4] imply that we can restrict ourselves to the case when G is a finite p-group. Let

1⟶Z⟶H⁢⟶𝜋⁢G⟶1

be a CP cover of G, where Z is a central subgroup of H with the property that Z≅B0⁡(G) and Z∩K⁡(H)=1. From here on, the proof goes on by considering each case separately.

3.1 Metabelian groups

The case of metabelian groups is easy:

Theorem 3.2.

Let G be a metabelian group of finite exponent. Then the exponent of G⋏G divides exp⁡G.

Proof.

Put exp⁡G=e. Note that H is also metabelian; hence it suffices to prove that [x,y]e=1 for all x,y∈H. We expand

1=[x,ye]=[x,y]e⁢∏k=2e[x,yk](ek).

Observe that ∏k=2e[x,yk](ek)∈Z. Furthermore,

∏k=2e[x,yk](ek)=[∏k=2e[x,yk-1](ek),y]∈K⁡(H);

therefore, [x,y]e=1, as required. ∎

3.3 Exponent 4

At first, we list some properties of groups of exponent 4 that will be used in the proof of this case.

Lemma 3.4.

Let G be a group of exponent 4 and a,b,c∈G. Then

the group 〈a,b〉 is nilpotent of class ≤5, 〈a,b,c〉 is nilpotent of class ≤7, and 〈[a,b],c〉 is nilpotent of class ≤4,
[[a,b]2,a]=1,
[a,b,a,a2⁢[a,b]]=1,
[c,[a,b],[a,b],[a,b]]=1.

Proof.

All the above properties can be deduced immediately from a polycyclic presentation of B⁢(3,4); see also [7]. ∎

Theorem 3.5.

Let G be a group of exponent 4. Then the exponent of G⋏G divides 4.

Proof.

As noted above, we may assume without loss of generality that G is a finite group. Choose x,y,z∈H.

First note that [[x,y]2,x]∈Z∩K⁡(H)=1 by Lemma 3.4; therefore,

(3.1)1=[x,y,x]2⁢[x,y,x,[x,y]].

Take w∈{x,y}. As 〈x,y〉 is nilpotent of class ≤5, we get

1=[[x,y]2,x,w]=[x,y,x,w]2.

From here, it follows that

(3.2)γ4⁢(〈x,y〉)2=1.

We also have that [x,y,z]4=1 by [14, Proof of Theorem 2.6]. Now we expand 1=[x4,y] using [15, Lemma 9]:

1=[x4,y]=[x,y]4⁢[x,y,x]6⁢[x,y,x,x]4⁢[x,y,x,x,x]⁢[x,y,x,[x,y]]14=[x,y]4⁢[x,y,x]2⁢[x,y,x,x,x].

Lemma 3.4 implies that [x,y,x,x2⁢[x,y]]=1. Expanding this using the class restriction, we obtain 1=[x,y,x,x]2⁢[x,y,x,x,x]⁢[x,y,x,[x,y]], and this implies [x,y,x,x,x]=[x,y,x,[x,y]]. From (3.1) and the above expansion, we get that [x,y]4=1.

By Lemma 3.4, the group 〈[x,y],z〉 is metabelian and nilpotent of class ≤4, and we also have that [z,[x,y],[x,y],[x,y]]=1. We now expand ([x,y]⁢z)4 using Subsection 2.2 and (3.2) as follows:

([x,y]⁢z)4=z4⁢[z,[x,y]]2⁢[z,[x,y],[x,y],z]⁢[z,[x,y],z,z].

Denote w=[z,[x,y]]2⁢[z,[x,y],[x,y],z]⁢[z,[x,y],z,z], and consider the following words:

w1=[x,z,z,x2⁢z2⁢[z,y,x]⁢[z,x,y,y]],

w2=[y2⁢z2,[z,y,z]],

w3=[y2⁢z2,[y,x,x,x]⁢[z,x,x,x]],

w4=[y2⁢z2,[z,x,z,z]⁢[z,y,y,x,x]],

w5=[[z,y][z,x]z2,z2[z,x][z,y][z,x,x][z,x,y][z,y,x][z,y,y]×[z,x,x,x][z,y,y,x,x]],

w6=[[z,y]⁢[z,x]⁢z2,[z,y,z,x]⁢[z,y,z,y]⁢[z,y,z,x,x]],

w7=[[z,y]⁢[z,x]⁢z2,[z,x,z,z]].

The subgroup 〈x,y,z〉 is an image of

K=〈a,b,c∣⁢class⁢ 7,laws⁢[x14,x2]=[x1,x2]4=[[x1,x2]2,x1]=[x1,3[x2,x3]]=1〉.

Expansions of the above defined words in K into products of basic commutators reveal that w=w1⁢w2⁢⋯⁢w7. On the other hand, inspection of the presentation of B⁢(3,4) shows that wi∈K⁡(H)∩Z=1 for all i=1,2,…,7; therefore, w=1. This immediately implies ([x,y]⁢z)4=z4 for all x,y,z∈H. From here, it is not difficult to conclude that exp⁡γ2⁢(H) divides 4, and this finishes the proof. ∎

3.6 4-Engel groups

The aim of this section is to prove the following:

Theorem 3.7.

Let G be a 4-Engel group of finite exponent. Then the exponent of G⋏G divides exp⁡G.

As 4-Engel groups are locally nilpotent [8], the situation can be easily reduced to the case when G is a finite p-group. If p≠2,5, then it follows from [16] that even exp⁡(G∧G) divides exp⁡G. This is no longer true when p=2 or p=5. In the case when p=5, there is a short proof of Theorem 3.7. Let H be a CP cover of G, and denote exp⁡G=5e. Note that H is a 4-Engel 5-group; hence it is regular [9, Remark 3.2]. It follows from [16, p. 1107] that if x,y∈H, then [x,y]5e=1. Regularity now implies that γ2⁢(H) has exponent dividing 5e.

We are thus left with 4-Engel 2-groups. The argument here is more involved. We start with some preliminaries.

Lemma 3.8.

Let G be a 4-Engel group of exponent 2e and a,b,c∈G. Then

γ7⁢(〈a,b〉)=γ8⁢(〈a,b,c〉)2=γ9⁢(〈a,b,c〉)=1,
[a,b,a]2e-1=[a,b,b]2e-1=1,
γ4⁢(〈a,b,c〉)2e-1=1.

Proof.

It follows from [19] that if 〈a,b,c〉 is a 4-Engel group, then

γ7⁢(〈a,b〉)=(γ8⁢(〈a,b,c〉)/γ9⁢(〈a,b,c〉))30=γ9⁢(〈a,b,c〉)3=1.

This proves (a). The fact that the exponent of γ4⁢(〈a,b,c〉) divides 2e-1 is proved in [16, Lemma 4.6], whereas the proof of that Lemma also yields (b). ∎

We will also use the following:

Lemma 3.9.

Let G be a 4-Engel group, a,b∈G and n a non-negative integer. Then

[an,b]=[a,b]n⁢[a,b,a](n2)⁢[a,b,a,a](n3)⁢[a,b,a,[a,b]](n2)+2⁢(n3).

Proof.

This follows from [16, Lemma 4.4], which gives a similar expansion of [an,b] in center-by-4-Engel groups by noting that the last two commutators of that lemma are trivial in 4-Engel groups. ∎

Referring to a polycyclic presentation of the free 4-Engel group with two or three generators obtained in [19], we have the following:

Lemma 3.10.

Let G be a 4-Engel group and a,b∈G. Then

[b,a,a,[b,a],a]=[b,a,a,a,b,a]=1,
[b,a,a,b,a]3⁢[b,a,b,b,a]=[a,b,a,[a,b]]⁢[b,a,b,a,b,a]3,
γ6⁢(〈a,b〉)=〈[b,a,b,a,b,a]〉,
if G has no elements of order 3 , then
[c,[a,b],[a,b],[a,b]]∈γ7⁢(〈a,b,c〉)2⁢γ8⁢(〈a,b,c〉).

Proposition 3.11.

Let G be 4-Engel group of exponent 2e. Then

[[a,b]2e-1,a]=1,
[c,[a,b],[a,b],[a,b]]2e-2=1.

Proof.

By Theorem 3.5, we may assume that e>2. Let us expand (a⁢b)2e=1 using Subsection 2.2 and Lemma 3.8. We obtain

(3.3)[a,b](2e2)=([b,a,a,a]⁢[b,a,a,b]⁢[b,a,b,b]⁢[b,a,a,[b,a]]⁢[b,a,a,a,b]×[b,a,b,[b,a,a]][b,a,a,a,b,b])(2e4).

We commute this with a and apply class restriction and Lemma 3.10 (a):

(3.4)[[a,b](2e2),a]=([b,a,a,b,a]⁢[b,a,b,b,a])(2e4).

Using Lemma 3.8 and Lemma 3.9, we obtain after a short calculation that

(3.5)[[a,b]2e-1,a]=[a,b,a,[a,b]](2e-12).

Equations (3.4) and (3.5), together with Lemma 3.10 (b), give

[b,a,b,a,b,a]2e-2=1.

This immediately yields γ6⁢(〈a,b〉)2e-2=1 by Lemma 3.10 (c). Now replace b by ab in (3.3) and use (3.3). Expansion under given class restriction gives

(3.6)1=([b,a,b,a]⁢[b,a,a,b])2e-2.

If we replace b by ab in (3.6) and apply (3.6), we obtain [b,a,a,b,a]2e-2=1. Replacing a by ba in this identity, we conclude that also [b,a,b,b,a]2e-2=1. Equation (3.4) now gives

[[a,b](2e2),a]=1.

This proves (a), whereas (b) follows directly from Lemma 3.10 (d) and Lemma 3.8 (a) and (c), as e>2. ∎

Theorem 3.12.

Let G be a 4-Engel group of exponent 2e. Then the exponent of G⋏G divides 2e.

Proof.

Note that H is a 4-Engel group. Take x,y,z∈H, and let a=xπ, b=yπ, c=zπ. Proposition 3.11 implies

1=[[x,y]2e-1,x]=[x,y,x]2e-1⁢[x,y,x,[x,y]](2e-12).

Equation (3.5) implies [a,b,a,[a,b]](2e-12)=1; therefore,

[x,y,x,[x,y]](2e-12)=[[x,y,x](2e-12),[x,y]]∈K⁡(H)∩Z=1.

This gives [x,y,x]2e-1=1. From Lemma 3.9, we get

1=[x2e,y]=[x,y]2e⁢[x,y,x](2e2)⁢[x,y,x,x](2e3)⁢[x,y,x,[x,y]](2e2)+2⁢(2e3).

Using the above equations, we see that this identity implies [x,y]2e=1. Now note that the subgroup 〈[x,y],z〉 is nilpotent of class ≤5 since H is 4-Engel. We expand ([x,y]⁢z)2e using the collection process (see Subsection 2.2):

(3.7)([x,y]z)2e=z2e([z,[x,y],[x,y],[x,y]]×[z,[x,y],[x,y],z][z,[x,y],z,z])(2e4).

Note that [z,[x,y],[x,y],[x,y]](2e4)=[[z,[x,y]](2e4),[x,y],[x,y]]∈K⁡(H), and Proposition 3.11 implies that

[z,[x,y],[x,y],[x,y]](2e4)∈Z.

This immediately shows that [z,[x,y],[x,y],[x,y]](2e4)=1. Thus

([z,[x,y],[x,y],z]⁢[z,[x,y],z,z])(2e4)∈Z.

Furthermore, the class restriction yields that

([z,[x,y],[x,y],z]⁢[z,[x,y],z,z])(2e4)=[[z,[x,y],[x,y]]⁢[z,[x,y],z],z](2e4)=[([z,[x,y],[x,y]]⁢[z,[x,y],z])(2e4),z]∈K⁡(H);

therefore, we conclude that ([z,[x,y],[x,y],z]⁢[z,[x,y],z,z])(2e4)=1. Thus equation (3.7) gives ([x,y]⁢z)2e=z2e, and induction on the commutator length shows that exp⁡H′ divides 2e. ∎

3.13 Groups of nilpotency class ≤5

We will prove the following:

Theorem 3.14.

Let G be a group of finite exponent and class ≤5. Then the exponent of G⋏G divides exp⁡G.

Again, we may assume that G is a finite p-group of class ≤5 and exponent pe. The CP cover H of G is then also nilpotent of class ≤5. Let x,y∈H. Assume first that p>2. Then it follows from the proof of [15, Theorem 13] that [x,y]pe=1. As [H,H] is nilpotent of class ≤2, it is regular; hence exp⁡[H,H] divides pe. Thus we are left with the case when G is a 2-group. Without loss, we can assume that e>2.

Take g,h,k∈G. Then the expansion of (g⁢[h,k])2e=1 yields

1=[h,k,g](2e2)⁢[h,k,g,[h,k]](2e2)+2⁢(2e3)⁢[h,k,g,g,g](2e4).

If we replace h by a commutator [h1,h2] in the above equation, we get, after renaming the variables, that

[h1,h2,h3,h4]2e-1=1;

therefore, γ4⁢(G)2e-1=1. By the class restriction, this implies γ4⁢(H)2e-1=1. Now take x,y,z∈H. Then

1=[[x,y]2e,z]=[x,y,z]2e⁢[x,y,z,[x,y]](2e2)=[x,y,z]2e;

hence γ3⁢(H)2e=1. As [H,H] is nilpotent of class ≤2, it suffices to prove that [x,y]2e=1 for all x,y∈H, and then Theorem 3.14 follows.

Take x,y∈H. Then

(3.8)1=[x2e,y]=[x,y]2e⁢[x,y,x](2e2)⁢[x,y,x,x,x](2e4).

If we interchange x and y in (3.8), we get

(3.9)1=[x,y]2e⁢[x,y,y](2e2)⁢[x,y,y,y,y](2e4).

Now we replace x by yx in (3.8) and apply (3.8) and (3.9). After a short calculation, we obtain

(3.10)[x,y]2e=([x,y,x,x,y]⁢[x,y,x,y,x]⁢[x,y,x,y,y]×[x,y,y,x,x][x,y,y,x,y][x,y,y,y,x])(2e4).

As H is nilpotent of class ≤5, we have that [x,y,x,y,x]=[x,y,y,x,x] and [x,y,x,y,y]=[x,y,y,x,y]. Thus (3.10) can be rewritten as

(3.11)[x,y]2e=([x,y,x,x,y]⁢[x,y,y,y,x])(2e4).

Denote f=(2e4), and

u=[y,x]⁢[y,x,y],

v=[y,x]-1⁢[y,x,y]-1⁢[y,x,x,x]⁢[y,x,y,x],

w=[yf⁢u,y-f⁢v].

We expand w:

w=[yf⁢u,v]⁢[yf⁢u,y-f]v

=[yf,v]u⁢[u,v]⁢[u,y-f]v

=[yf,v]⁢[yf,v,u]⁢[u,v]⁢[u,y-f]⁢[u,y-f,v]

=[yf,v]⁢[u,y-f]⁢[u,v]⁢([y,v,u]⁢[u,y,v]-1)f.

Note that

[u,v]=[[y,x],[y,x,y]-1]⁢[[y,x,y],[y,x]-1]=[[y,x],[y,x,y]]-1⁢[[y,x,y],[y,x]]-1=1,

and the Hall–Witt identity gives 1=[y,v,u]⁢[v,u,y]⁢[u,y,v]=[y,v,u]⁢[u,y,v]. Thus w=[yf,v]⁢[u,y-f]. Now

[u,y-f]=[(y-1)f,u]-1=[y-1,u]-f⁢[y-1,u,y-1]-(f2)⁢[y-1,u,y-1,y-1]-(f3).

Quick calculation shows that

[y-1,u]=[y,x,y],

[y-1,u,y-1]=[y,x,y,y,y]⁢[y,x,y,y]-1,

[y-1,u,y-1,y-1]=[y,x,y,y,y].

Therefore,

[u,y-f]=[y,x,y]-f⁢[y,x,y,y](f2)⁢[y,x,y,y,y]-(f2)-(f3).

On the other hand, we easily get

[y,v]=[y,x,y]⁢[y,x,y,[x,y]]⁢[y,x,y,y]⁢[y,x,x,x,y]⁢[y,x,y,x,y],

and thus

[yf,v]=[y,v]f⁢[y,v,y](f2)⁢[y,v,y,y](f3)=[y,x,y]f⁢[y,x,y,[x,y]]f⁢[y,x,y,y]f+(f2)×[y,x,x,x,y]f⁢[y,x,y,x,y]f⁢[y,x,y,y,y](f2)+(f3).

We thus get

w=[y,x,y,y]f+2⁢(f2)⁢[y,x,y,[x,y]]f⁢([y,x,x,x,y]⁢[y,x,y,x,y])f.

Note that

[y,x,y,y]f+2⁢(f2)=[y,x,y,y]f2=1

since f2 is divisible by 22⁢e-4≥2e-1. As H has class ≤5, we also have

[y,x,y,[x,y]]=[y,x,y,x,y]⁢[x,y,y,y,x].

This, together with (3.11), implies w=([x,y,x,x,y]⁢[x,y,y,y,x])f=[x,y]2e. We conclude that [x,y]2e∈K⁡(H)∩Z=1, and this proves Theorem 3.14.

Communicated by Andrea Lucchini

Funding statement: The author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0222, and projects No. J1-8132, J1-7256 and N1-0061).

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Received: 2018-03-24

Revised: 2018-11-22

Published Online: 2019-02-15

Published in Print: 2019-05-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2018-0041