1 Introduction
In 1934, Reinhold Baer [1] defined the normN(G) of a group G to be the intersection of the normalizers of all subgroups of G. Of course, the norm of any group G contains the centre Z(G), and a group coincides with its norm if and only if all its subgroups are normal. It was proved by Baer that N(G) is abelian if G is not periodic and that N(G)=Z(G) whenever N(G) contains elements of infinite order. Later, Schenkman [21] showed that N(G) is always contained in the second term ζ2(G) of the upper central series of G. This latter result is also a direct consequence of Cooper’s theorem on centrality of power automorphisms (see [4]), since an element belongs to the norm of a group if and only if it acts by conjugation as a power automorphism. Notice also that Beidleman, Heineken and Newell [3] proved that if G is a primary group, then at least one of the groups N(G)/Z(G) and [N(G),G] is cyclic.
The concept of norm was recently generalized by restricting attention just to normalizers of certain subgroups (see [2]). In fact, if 𝔛 is any group class, the 𝔛-norm of a group G is the intersection of all normalizers of subgroups of G which are not in 𝔛. In particular, if ℑ is the class consisting only of trivial groups, the ℑ-norm coincides with the norm introduced by Baer. For instance, this approach has been adopted in [8], where the intersection of all normalizers of non-periodic subgroups has been studied (this is of course the 𝔗-norm, where 𝔗 is the class of all periodic groups). For an updated report on properties of the various relevant norms of a group, see the recent survey paper [12].
A group G is called metahamiltonian if every non-abelian subgroup of G is normal. Metahamiltonian groups were introduced and studied in a series of papers by Romalis and Sesekin (see [18, 19, 20]), who proved in particular that the commutator subgroup of any soluble group G with such property is finite and has prime-power order. It follows that torsion-free soluble metahamiltonian groups are abelian, and that the second term of the upper central series of any soluble metahamiltonian group has finite index. Of course, any group whose proper subgroups are abelian is metahamiltonian, so in particular Tarski groups, i.e. infinite simple groups whose proper non-trivial subgroups have prime order, belong to the class of metahamiltonian groups. In order to avoid them and other similar pathologies some weak solubility requirement is necessary. Actually, the results of Romalis and Sesekin can be proved within the wide universe of locally graded groups (see [9]). Recall that a group G is locally graded if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index; in particular, all locally (soluble-by-finite) groups are locally graded. Further relevant properties of metahamiltonian groups can be found in [6, 7, 13].
The metanorm of a group G is the subgroup M(G) consisting of all elements g such that Xg=X for each non-abelian subgroup X of G. Thus M(G) is the intersection of the normalizers of all non-abelian subgroups of G, i.e. the 𝔄-norm of G, where 𝔄 is the class of all abelian groups. Clearly, M(G) is a metahamiltonian characteristic subgroup of G, and a group is metahamiltonian if and only if coincides with its metanorm. The influence of the behaviour of the metanorm on the structure of the whole group has been investigated in [5]; for instance, it turns out that if G is any locally finite group whose metanorm is metabelian but not nilpotent, then G is metahamiltonian (or equivalently M(G)=G), unless the order of the commutator subgroup of M(G) is the square of a prime number.
The aim of this paper is to study embedding properties of the metanorm. In particular, it will be shown that the subgroup [M(G),G] is periodic for any locally graded group G, and the behaviour of the metanorm in locally nilpotent groups will be studied in detail. Moreover, it will be proved that if G is any locally graded group with a non-nilpotent metanorm, then the only obstruction to a strong hypercentral embedding of M(G) in G is represented by the section M(G)′/M(G)′′, which is a small eccentric chief factor. In fact, in this situation it turns out that the subgroup M(G)′′ is contained in the centre of G and M(G)/M(G)′ lies in the centre of G/M(G)′. As a consequence, the second term ζ2(M(G)) of the upper central series of M(G) is contained in ζ2(G).
Most of our notation is standard and can be found in [17].
2 Some general results
Let G be a locally graded metahamiltonian group which is not abelian. It follows from the results of Romalis and Sesekin that the commutator subgroup G′ of G is a finite p-group for some prime number p, which will be called here the characteristic of G. The structure of metahamiltonian groups was described in details by Kuzennyi and Semko; they proved in particular the following two results which are essential for our purposes (see [13]).
Lemma 2.1.
Let G be a metahamiltonian group which is metabelian but not nilpotent, and let p be the characteristic of G. Then G′ is a minimal normal subgroup of G and the unique Sylow p-subgroup of G is abelian. Moreover, there exists a subgroup K such that G=G′K and G′∩K={1}.
Lemma 2.2.
Let G be a metahamiltonian locally graded group which is not metabelian, and let p be the characteristic of G. Then G is periodic, G′ has order p3, G′′ is a subgroup of order p of Z(G) and G/G′ has no elements of order p. Moreover, G′/G′′ is an eccentric chief factor of G and the factor group G/CG(G′/G′′) is cyclic.
It follows from the above results that if G is any locally graded metahamiltonian group which is not nilpotent, then G′/G′′ is a chief factor of G, which must be eccentric since G/G′′ cannot be nilpotent by Philip Hall’s nilpotency criterion. Notice also that nilpotent metahamiltonian groups are metabelian.
A celebrated theorem of Issai Schur shows that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite (see [17, Part 1, Theorem 4.12]). As for any group G the section N(G)/Z(G) cannot be too large, it follows easily that the same conclusion holds when the norm of a group has finite index. This is also true if the norm is replaced by the metanorm, at least in the locally graded case.
Lemma 2.3.
Let G be a locally graded group whose metanorm M(G) has finite index. Then the commutator subgroup G′ of G is finite.
Proof.
As the index |G:M(G)| is finite, G admits only finitely many normalizers of non-abelian subgroups, and then it is well known that the subgroup G′ is finite (see [10, Theorem A]).
∎
Recall that a group G is said to be residually finite if the intersection of all its (normal) subgroups of finite index is trivial. In particular, every free group is residually finite and all residually finite groups are locally graded.
Lemma 2.4.
Let G be a group and M=M(G) the metanorm of G. If X is a residually finite subgroup of G which is not abelian-by-finite, then [M,X,X]={1}.
Proof.
Let Y be any subgroup of finite index of X. Then Y is not abelian, and so all subgroups of G containing Y are normalized by M. In particular, M induces a group of power automorphisms on X/Y. Since any power automorphism is a central automorphism, it follows that [M,X,X]≤Y, and so [M,X,X]={1} since X is residually finite.
∎
It is known that if G is a group and the factor group G/Z(G) is polycyclic-by-finite, then G′ is also polycyclic-by-finite (see [17, Part 1, p. 115]), a result which corresponds to Schur’s theorem quoted above when finiteness is replaced by the property of being polycyclic-by-finite. Our next lemma shows that a similar statement holds for the metanorm of a locally graded group, and will be used in Section 4.
Lemma 2.5.
Let G be a locally graded group such that the factor group G/M(G) is polycyclic-by-finite. Then also the commutator subgroup G′ of G is polycyclic-by-finite.
Proof.
As the commutator subgroup M′ of M=M(G) is finite, the group G is soluble-by-finite, and so the largest soluble normal subgroup S of G has finite index. An obvious induction on the derived length of S yields that S′′ is finite, so G may be replaced by G/S′′ and hence without loss of generality it can be assumed that G is metabelian-by-finite. Let E be a finitely generated non-abelian subgroup of G such that G=EM. Then E is normal in G and the factor group G/EM′ is abelian, so G′ is contained in EM′. Since E is residually finite (see for instance [17, Part 2, Theorem 9.51]), by Lemma 2.4 we have that either E is abelian-by-finite or [M,E,E]={1}. In the latter case M∩E is contained in ζ2(E), so E/ζ2(E) is polycyclic-by-finite and hence also γ3(E) is polycylic-by-finite (see [17, Part 1, p. 113]). Thus E is polycyclic-by-finite in any case, and hence G′ is polycyclic-by-finite because M′ is finite.
∎
Recall that any group G contains a largest locally nilpotent normal subgroup, which is called the Hirsch–Plotkin radical of G and contains all ascendant locally nilpotent subgroups. A group G is said to be radical if it has an ascending (normal) series with locally nilpotent factors, or equivalently if the upper Hirsch–Plotkin series (i.e. the series of iterated Hirsch–Plotkin radicals) reaches G. Thus the subgroup generated by any collection of radical normal subgroups is likewise radical, and every radical normal subgroup of a group G is contained in the last term of the upper Hirsch–Plotkin series of G. Recall also that a group G is said to be an 𝐹𝐶-group if each element of G has only finitely many conjugates. Of course, abelian groups and finite groups have the 𝐹𝐶-property, and it is well known that the commutator subgroup of an 𝐹𝐶-group is locally finite (see for instance [17, Part 1, Theorem 4.32]). Moreover, any group with finite commutator subgroup has boundedly finite conjugacy classes, and so in particular all metahamiltonian locally graded groups have the 𝐹𝐶-property.
Notice that Ol’shanskiĭ constructed a simple group whose proper non-trivial subgroups are infinite cyclic (see [15, Theorem 28.3]), so there exist perfect torsion-free metahamiltonian groups. On the other hand, the main result of this section shows that the subgroup [M(G),G] cannot contain elements of infinite order, for any locally graded group G.
Theorem 2.6.
Let G be a locally graded group, and let M=M(G) be its metanorm. Then the subgroup [M,G] is periodic.
Proof.
Let T be the largest periodic radical normal subgroup of G. Then T is contained in the last term of the upper Hirsch–Plotkin series of G, and so the factor group G/T is locally graded (see [14]). Moreover, G/T has no periodic radical non-trivial normal subgroups, and so in particular its metanorm is torsion-free. Since it is clearly enough to show that the statement holds for G/T, it can be assumed without loss of generality that the metahamiltonian subgroup M is torsion-free, and hence also abelian.
Assume for a contradiction that M is not contained in Z(G), so that there exist elements a∈M and x∈G such that [a,x]≠1. Put A=〈a〉〈x〉, so that 〈x,A〉=〈x,a〉. Suppose first that A∩〈x〉={1}. As 〈x,A〉 is a finitely generated metabelian group, it is residually finite (see [17, Part 2, Theorem 9.51]), so in particular
⋂n>0An={1}
and hence
⋂n>0〈x,An〉=〈x〉.
Clearly, the subgroup 〈x〉 is not normal in 〈x,a〉, and so there is a positive integer k such that 〈x,Ak〉 is abelian. Then
[a,x]k=[ak,x]=1 and so [a,x]=1, since A is torsion-free. This contradiction shows that
A∩〈x〉≠{1}.
Then A has finite index in 〈x,A〉, and hence it is finitely generated. Let m be the order of the subgroup consisting of all elements of finite order of A/A∩〈x〉. Then
〈x,Am〉/A∩〈x〉=〈x¯〉⋉B¯,
where x¯=x(A∩〈x〉) has finite order and
B¯=Am(A∩〈x〉)/(A∩〈x〉)
is torsion-free.
Since the non-abelian group 〈x,A〉 has torsion-free commutator subgroup, it is not an 𝐹𝐶-group and so x has infinitely many conjugates in 〈x,A〉. It follows that 〈x〉 is not normal in 〈x,Am〉, and hence 〈x¯〉 is not normal in 〈x¯,B¯〉. Thus [B¯,x¯]≠{1} and so also [B¯p,x¯]≠{1} for each prime number p. On the other hand, B¯ is contained in the metanorm M(〈x¯,B¯〉) of 〈x¯,B¯〉, and hence the non-abelian subgroup 〈x¯,B¯p〉 is normal in 〈x¯,B¯〉 for all p. Therefore
〈x¯〉=⋂p〈x¯,B¯p〉
is likewise a normal subgroup of 〈x¯,B¯〉, and this last contradiction completes the proof.
∎
Corollary 2.7.
Let G be a locally graded group, and let N be a torsion-free G-invariant subgroup of M(G). Then N is contained in the centre of G.
Proof.
The subgroup [N,G] is periodic by Theorem 2.6, and so it is trivial since N is torsion-free. Therefore N lies in the centre of G.
∎
3 Locally nilpotent groups
The aim of this section is to find conditions under which the metanorm of a locally nilpotent group G is contained in the hypercentre of G, i.e. in the last term of the upper central series of G.
Notice that our next lemma holds in particular when A is a p-subgroup for some prime number p and g is any element of G whose order is a power of p.
Lemma 3.1.
Let G be a group, and let A be an abelian G-invariant subgroup of M(G) and g an element of G such that 〈g,A〉 is locally nilpotent. Then there exist elements a and b of A such that [a,g]=1, [b,g]=a and [A,g] is contained in 〈a,b〉(〈g〉∩A). In particular, if A is periodic, the subgroup [A,g] is finite.
Proof.
Obviously, it can be assumed that [A,g]≠{1}. Since 〈A,g〉 is locally nilpotent, there exist non-trivial elements a1,…,ak of A such that
[a1,g]=1,[a2,g]=a1,…,[ak,g]=ak-1.
Then 〈a2,g〉 is a non-abelian subgroup, so it is fixed by each element of A, and hence [u,g] belongs to 〈a2,g〉 for all u∈A. It follows that [A,g] is contained in
〈a2,g〉∩A=(〈a1,a2〉)(〈g〉∩A),
and the statement is proved.
∎
If A is a periodic abelian group, we shall denote by Ω1(A) the socle of A, which is of course the subgroup consisting of all elements of square-free order. The upper socle series
{1}=Ω0(A)≤Ω1(A)≤…≤Ωn(A)≤Ωn+1(A)≤⋯
is defined by putting
Ωn+1(A)/Ωn(A)=Ω1(A/Ωn(A))
for each positive integer n.
Lemma 3.2.
Let G be a locally nilpotent group , and let A be a periodic abelian G-invariant subgroup of M(G). If [Ω1(A)∩G′,G′]={1}, then Ω1(A) is contained in the term ζ5(G) of the upper central series of G.
Proof.
Clearly, it is enough to show that each primary component of Ω1(A) lies in ζ5(G), and so without loss of generality it can be assumed that A is a p-group for some prime number p. Put B=Ω1(A)∩G′, and suppose that CG(B) is a proper subgroup of G. Let g be any element of G∖CG(B). Since G is locally nilpotent, it follows from Lemma 3.1 that [B,g] is finite, of order at most p3. Consider the map
φ:B→[B,g],
defined by putting bφ=[b,g] for each b∈B. Clearly, φ is a homomorphism whose kernel
CB(g)=CB(〈g,CG(B)〉)
is a normal subgroup of G, so B/CB(g) is a normal section of G of order at most p3. It follows that [B,G,G,G] lies in CG(g), so
[B,G,G,G,G]={1}
and hence B is contained in ζ4(G). On the other hand, Ω1(A)/B is obviously contained in the centre of G/B, and so Ω1(A) lies in ζ5(G).
∎
Corollary 3.3.
Let G be a locally nilpotent group, and let A be a periodic abelian G-invariant subgroup of M(G). If [A∩G′,G′]={1}, then A is contained in ζω(G), where ω is the first transfinite ordinal.
Proof.
It follows from Lemma 3.2 that Ωn+1(A)/Ωn(A) lies in ζ5(G/Ωn(A)) for all n, so
A=⋃n∈ℕΩn(A)
is contained in ζω(G).
∎
Let G be a locally nilpotent group. We now show that the only obstruction to M(G) being hypercentrally embedded in G is eventually caused by M(G)∩G′′.
Theorem 3.4.
Let G be a locally nilpotent group. Then M(G)/M(G)∩G′′ is contained in ζω+1(G/M(G)∩G′′).
Proof.
Put M=M(G), and consider the group G¯=G/M∩G′′. Then the subgroup M¯=M/M∩G′′ is contained in the metanorm of G¯, and H¯=[M¯,G¯] is periodic by Theorem 2.6. Clearly, [H,G′] lies in M∩G′′, so [H¯,G′¯]={1} and it follows from Corollary 3.3 that H¯/H¯′ is contained in ζω(G¯/H¯′). On the other hand, as H¯′ is finite, there is a positive integer k such that H¯′≤ζk(G¯) and hence H¯ is a subgroup of ζω(G¯). Finally, M¯/H¯ lies in Z(G¯/H¯), and so M¯ is contained in ζω+1(G¯).
∎
Theorem 3.4 has the following consequence, showing in particular that the metanorm of a metabelian locally nilpotent group G is always contained in the subgroup ζω+1(G).
Corollary 3.5.
Let G be a locally nilpotent group. If M(G)∩G′′ is finite, then M(G) is contained in ζω+1(G).
Proof.
As M(G)∩G′′ is finite, it is contained in ζk(G) for some positive integer k, and so M(G) lies in ζω+1(G) by Theorem 3.4.
∎
Corollary 3.6.
Let G be a locally nilpotent group such that G′ is contained in M(G). Then M(G) lies in ζω+1(G) and so G is hypercentral of length at most ω+2.
Proof.
As G′ is contained in the metahamiltonian group M(G), its commutator subgroup G′′ is finite. Then it follows from Corollary 3.5 that M(G) lies in ζω+1(G), and hence G=ζω+2(G).
∎
We leave here as an open question whether the statement of Corollary 3.5 can be improved by showing that, under the same assumptions, the metanorm M(G) lies in ζω(G). On the other hand, it is well known that for an arbitrary group G, all elements of ζω(G) are bounded right Engel elements (see for instance [17, Part 2, Lemma 7.12]), and our next result shows that the elements of the metanorm of a locally nilpotent group behave similarly.
Corollary 3.7.
Let G be a locally nilpotent group whose metanorm M(G) is abelian. Then every element of M(G) is right 3-Engel in G.
Proof.
Let x be any element of M=M(G), and consider an arbitrary element g of G. It follows from Lemma 3.1 that there exist elements y and z of M such that [y,g]=1, [z,g]=y and the commutator [x,g] belongs to 〈y,z〉(〈g〉∩M). Thus [x,g]=yhzkal, where 〈a〉=〈g〉∩M and h,k,l are suitable positive integers, so
[x,g,g]=[yhzkal,g]=[zk,g]=yk
and hence
[x,g,g,g]=[yk,g]=1.
Therefore x is a right 3-Engel element of G.
∎
Lemma 3.8.
Let G be a locally nilpotent group, and let A be a periodic abelian G-invariant subgroup of M(G). If G/CG(A) is finite, then A is contained in ζm(G) for some positive integer m.
Proof.
As G/CG(A) is finite, there exist elements g1,…,gt such that
G=〈CG(A),g1,…,gt〉.
Moreover, it follows from Lemma 3.1 that the subgroup [A,gi] is finite for each i=1,…,t, so the index |A:CA(gi)| is finite and hence also the intersection
C=⋂i=1tCA(gi)
has finite index in A. Clearly, C is contained in the centre of G, and A/C is a finite normal subgroup of G/C, so A/C is contained in ζn(G/C) for some positive integer n and hence A lies in ζn+1(G).
∎
The last theorem of this section deals with the behaviour of the metanorm in a locally nilpotent group containing a nilpotent subgroup of finite index.
Theorem 3.9.
Let G be a locally nilpotent group. If G contains a nilpotent subgroup of finite index, then M(G) is contained in ζm(G) for some integer m>0.
Proof.
Let N be a nilpotent normal subgroup of G such that the index |G:N| is finite and the nilpotency class c of N is smallest possible. The statement is obvious if c=0, since in this case G is finite, and so also nilpotent. Suppose c>0, and put A=Z(N). It can be assumed by induction on c that the metanorm M(G/A) is contained in some term with finite ordinal type of the upper central series of G/A, and hence in particular
[M(G),G,…,G⏟h]≤A∩M(G)
for some positive integer h.
Put B=A∩M(G), and let T be the subgroup consisting of all elements of finite order of B. As N is contained in the centralizer CG(T), it follows that the index |G:CG(T)| is finite, and so by Lemma 3.8 there exists a positive integer k such that T lies in ζk(G). Finally, it follows from Lemma 2.7 that B/T is contained in the centre of G/T, and hence M(G)≤ζh+k+1(G).
∎
4 Groups with a non-nilpotent metanorm
The aim of this section is to study embedding properties of the metanorm M=M(G) of a locally graded group G, when M is not nilpotent. In this case, the description of the structure of metahamiltonian non-nilpotent groups given in Lemmas 2.1 and 2.2 can be used, so that in particular the finite group M′/M′′ is an eccentric chief factor of M, and so also of G. It will be proved that in this situation M′′ and M/M′ are central sections of G, so M is close to being hypercentrally embedded in G, and the subgroup ζ2(M) (which has finite index in M) is contained in ζ2(G).
Lemma 4.1.
Let G be a group, and let X be a non-abelian subgroup of G. If H and K are G-invariant subgroups of M(G) such that K≤H and H∩X≤K, then X centralizes H/K.
Proof.
As the subgroup X is not abelian, it is normalized by H, and hence
[H,X]≤H∩X≤K.
Therefore X centralizes the section H/K.
∎
We can now prove the first main theorem of this section.
Theorem 4.2.
Let G be a locally graded group, and let M=M(G) be the metanorm of G. Then the subgroup M′′ is contained in the centre of G.
Proof.
Of course, it can be assumed that M is not metabelian, so that it follows from Lemma 2.2 that M is periodic, M′′ has prime order p, where p is the characteristic of M, M′ has order p3, M′/M′′ is an eccentric chief factor of M and M/CM(M′/M′′) is cyclic. Moreover, M/M′ is a p′-group, and so M′ is the largest p-subgroup of M.
Let a be any element of infinite order of G. Assume for a contradiction that [M′′,a]≠{1}. Clearly, the set π of all prime numbers q such that [M′′,aq]≠{1} is infinite. If q∈π, the subgroup 〈aq,M′′〉 is not abelian, so all subgroups of G containing 〈aq,M′′〉 are normalized by M. It follows that M〈aq,M′′〉/〈aq,M′′〉 is contained in the norm of 〈a,M〉/〈aq,M′′〉, and so also in the second centre of M〈aq,M′′〉/〈aq,M′′〉. Thus 〈a,M〉/〈aq,M′′〉 is nilpotent of class at most 2, and so
γ3(〈a,M〉)≤⋂q∈π〈aq,M′′〉=M′′.
In particular, M/M′′ is nilpotent, which is impossible, as it admits a non-central chief factor. Therefore [M′′,a]={1} for each element of infinite order a of G.
Consider now an arbitrary element of finite order g of G, and assume for a contradiction that [M′′,g]≠{1}. In order to complete the proof, the group G can be replaced by its locally finite subgroup 〈g,M〉, and so we may suppose without loss of generality that G is locally finite. Obviously p>2 and the factor group G/CG(M′′) is cyclic of order dividing p-1, so there is a non-trivial element x of order prime to p such that G=〈x〉CG(M′′). As 〈x,M〉/M′ is a p′-group, M′ is the unique Sylow p-subgroup of X=〈x,M〉, and hence there exists a p′-subgroup Q such that X=QM′. But M′ is finite, so its complements in X are pairwise conjugate (see for instance [11, Theorem 2.4.5]), and so the subgroup Q can be chosen containing x. Moreover, since [M′′,Q]≠{1}, it follows from Lemma 4.1 that Q is abelian. Obviously, M=Q0M′, where Q0=Q∩M, so M′/M′′ is an eccentric Q0-chief factor and Q0/CQ0(M′/M′′) is cyclic. Let y be an element such that
Q0=〈y〉CQ0(M′/M′′),
and consider the finite subgroup E=〈x,y〉 of Q. Then
M′=[M′,y]M′′≤[M′,E]M′′,
and so M′=[M′,E]M′′. On the other hand, the subgroup EM′′ is not abelian, so it is normalized by M, and hence in particular
M′=[M′,E]M′′≤M′∩EM′′=M′′.
This contradiction shows that [M′′,g]={1} for all g, and so M′′ is contained in the centre of G.
∎
In order to prove that [M(G),G]≤M(G)′, whenever G is a locally graded group and M(G) is not nilpotent, we need some further information on the behaviour of the metanorm.
It was shown by Kuzennyi and Semko [13] that in a locally graded metahamiltonian group G all non-abelian subgroups contain G′. In the case of groups with a non-nilpotent metanorm, this result can be extended in the following way.
Lemma 4.3.
Let G be a locally graded group whose metanorm M=M(G) is not nilpotent. Then every non-abelian subgroup of G contains M′.
Proof.
Assume for a contradiction that there exists a non-abelian subgroup X of G such that M′ is not contained in XM′′. As M normalizes X, the intersection XM′′∩M′ is normal in M, and so XM′′∩M′=M′′, since M′/M′′ is a chief factor of M. Thus M′∩X is contained in M′′. Moreover, M normalizes all subgroups of G containing X, and so in particular MX/X is a Dedekind group. It follows that
M′/M′∩X≃M′X/X=(MX/X)′
has order at most 2, so M′/M′′ lies in Z(M/M′′) and M is nilpotent. This contradiction proves that M′ is a subgroup of XM′′. As M′′ lies in Z(G) by Theorem 4.2, it follows that X′=(XM′′)′ contains M′′ and hence M′ lies in X.
∎
Corollary 4.4.
Let G be a locally graded group whose metanorm M=M(G) is not nilpotent, and let p be the characteristic of M. Then every subgroup of G with no elements of order p is abelian.
Lemma 4.5.
Let G be a locally graded group, and let X be a non-nilpotent subgroup of the metanorm M=M(G). Then X′=M′.
Proof.
As the metanorm M of G is not nilpotent, its commutator subgroup M′ is contained in X by Lemma 4.3. Then X′′≤M′′≤X′, and so X′′=M′′ since X′/X′′ is a chief factor of X and M′′≤Z(G). As M normalizes X and M′/M′′ is a chief factor of M, it follows that X′/M′′=M′/M′′ and hence X′=M′.
∎
Lemma 4.6.
Let G be a locally finite group whose metanorm M=M(G) is metabelian but not nilpotent. If p is the characteristic of M and P is any Sylow p-subgroup of G, then M′ is contained in Z(P).
Proof.
Clearly, M′ is contained in P, because it is a normal p-subgroup of G. Thus we may suppose that P is not abelian, so that M normalizes P, and hence also Z(P). It follows that M′∩Z(P) is a normal subgroup of M, which is not trivial since M′ is finite. On the other hand, M′ is a minimal normal subgroup of M by Lemma 2.1, so M′∩Z(P)=M′ and M′ lies in Z(P).
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Lemma 4.7.
Let G be a locally finite group whose metanorm M=M(G) is not nilpotent, and let p be the characteristic of G. If the commutator subgroup M′ of M is abelian and |M′|≠p, then the centralizer CG(M′) is locally nilpotent.
Proof.
Assume for a contradiction that the statement is false, and let E be a finite non-nilpotent subgroup of CG(M′) of smallest possible order. Then E is a minimal non-nilpotent group, so its order is divisible exactly by two prime numbers. On the other hand, M′ is contained in E by Lemma 4.3, so p divides the order of E and hence |E|=pmqn, where m,n are positive integers and q is a prime ≠p. Let P be a Sylow p-subgroup of E. As M′ is contained in P, we have that P cannot be cyclic and hence E=〈a〉⋉P for a suitable element a of order a power of q. Moreover,
M′≤P∩Z(E)≤Z(P),
so |Z(P)|>p and it follows from Rédei’s characterization of finite minimal non-nilpotent groups that P is a minimal normal subgroup of E (see [16]). Therefore P=M′, which is impossible, since [P,E]=P. This contradiction proves that the centralizer CG(M′) is locally nilpotent.
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Our next statement is the special case of the last main theorem, when the metanorm is not metabelian.
Lemma 4.8.
Let G be a locally graded group whose metanorm M=M(G) is not metabelian. Then M/M′ is contained in the centre of G/M′.
Proof.
The metahamiltonian group M is not metabelian, and so Lemma 2.2 yields that M is periodic, M′′ has prime order p, where p is the characteristic of M, M′ has order p3, M′/M′′ is an eccentric chief factor of M and M/CM(M′/M′′) is cyclic. Moreover, M/M′ is a p′-group, and hence M′ is the largest p-subgroup of M.
Let a be any element of infinite order of G. As M′ is not abelian, M/M′ is contained in the norm of the group 〈a,M〉/M′, which is generated by its elements of infinite order, so M/M′ lies in the centre of 〈a,M〉/M′ and hence [M,a]≤M′.
Consider now an arbitrary element of finite order g of G. Of course, M is contained in the metanorm of H=〈g,M〉, and it follows from Lemma 4.5 that M′=M(H)′, so in order to complete the proof the group G can be replaced by its locally finite subgroup H, and hence we may suppose without loss of generality that G is locally finite.
Let x be any element of G whose order is a power of p. As the p-subgroup P=〈x,M′〉 is not abelian, it is normalized by M and hence [P,M]≤P. Then the order of [P,M] is a power of p, and so [P,M] is contained in the largest p-subgroup M′ of M. Thus x acts trivially on M/M′. Consider now an element y of G whose order is prime to p. Then M′ is the unique Sylow p-subgroup of Y=〈y,M〉, and so there exists a p′-subgroup Q such that Y=QM′. Moreover, Q is abelian by Corollary 4.4, so also the group Y/M′≃Q is abelian, and hence y centralizes M/M′. Therefore M/M′ is contained in the centre of G/M′.
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Finally, we need the following lemma which has been proved in [5].
Lemma 4.9.
Let G be a locally finite group whose metanorm M=M(G) is metabelian but not nilpotent, and let p be the characteristic of M. Then either G is metahamiltonian or |M′|=p2.
Theorem 4.10.
Let G be a locally graded group whose metanorm M=M(G) is not nilpotent. Then M/M′ is contained in the centre of G/M′.
Proof.
By Lemma 4.8 it can be assumed that the metahamiltonian group M is metabelian. Suppose first that G is locally finite, and let p be the characteristic of M. If |M′|≠p2, it follows from Lemma 4.9 that G is metahamiltonian, so M=G and the statement is obvious in this case. Therefore it can also be assumed that M′ has order p2. Write
M/M′=P/M′×Q/M′,
where P/M′ is a p-group and Q/M′ is a p′-group. Assume for a contradiction that M/M′ is not contained in Z(G/M′), so there exist an element x of M of prime-power order qn and an element g of G such that [x,g] is not in M′. Let g=hk, where h has p-power order and the order of k is prime to p, and put H=〈x,k,M′〉.
Suppose that q≠p, so that x belongs to Q. Then H/M′ is a finite p′-group, and so H splits over M′ by the Schur-Zassenhaus theorem. It follows that H/M′ is isomorphic to a p′-subgroup of G, and hence it is abelian by Corollary 4.4. Therefore [x,k]∈M′, and so the element [x,h] does not belong to M′. Since M is not nilpotent, the set Q∖CQ(M′) contains an element y whose order is prime to p. Consider the finite non-abelian group
H/M′=〈xM′,hM′,yM′〉,
and its normal subgroup
K/M′=Q/M′∩H/M′.
Since K/M′ has order prime to p, there exists a subgroup L such that K=M′L and M′∩L={1}. Clearly, L is a Hall subgroup of K, and so an application of the Frattini argument to the finite soluble group H yields that H=KNH(L). It follows that H=M′NH(L), so the normalizer NH(L) cannot be abelian and hence it contains M′ by Lemma 4.3. Thus L is a normal subgroup of H=NH(L), and so K=M′×L, which is impossible because y lies in K. This contradiction proves that q=p, so x is an element of P and g does not belong to CG(P/M′).
By Lemma 4.6 the subgroup M′ centralizes all p-subgroups of G, and so in particular [M′,h]={1}. If g does not centralize M′, it follows that [M′,k]≠{1}, and hence P normalizes the non-abelian subgroup 〈k,M′〉; in this situation k acts trivially on P/M′, and so h cannot centralize P/M′. We have shown that in any case the set CG(M′)∖CG(P/M′) is non-empty. On the other hand, the centralizer CG(M′) is locally nilpotent by Lemma 4.7 and contains P by Lemma 4.6, and hence CG(M′)∖CG(P/M′) must contain an element c whose order is a power of p. Let z be an element of P such that [z,c] is not in M′. Then F=〈z,c〉 is a finite non-abelian p-group, and M′≤Z(F) by Lemma 4.3 and Lemma 4.6. Clearly, the intersection
P1/M′=P/M′∩F/M′
is a non-trivial normal subgroup of F/M′, and hence
P2/M′=P/M′∩Z(F/M′)≠{1}.
Moreover, P1/P2 is a non-trivial normal subgroup of F/P2, so we also have
P1/P2∩Z(F/P2)≠{1},
and in this latter intersection we may consider an element uP2 of order p. It follows from Lemma 2.1 that there exists a subgroup Y of M such that M=YM′ and Y∩M′={1}. Thus P=U×M′, where U=Y∩P, and the representative u can obviously be chosen in U. Since the coset uM′ is not in the centre of F/M′, there exists an element v of F such that w=[u,v] does not belong to M′. Thus the subgroup W=〈u,v〉 is not abelian, so it is normalized by M and contains M′. As the coset uM′ belongs to ζ2(F/M′), the factor group W/M′ is nilpotent of class 2, and so γ3(W) is an M-invariant subgroup of M′; in particular γ3(W) lies in Z(F). Moreover, [u,w]=1, because P is abelian, and so
γ3(W)=〈[v,w]〉W=〈[v,w]〉
is cyclic. On the other hand, M′ is a minimal normal subgroup of M, and hence γ3(W)={1}, so W′=〈w〉, and 〈w〉 is normalized by M. Then M normalizes also the subgroup 〈u,w〉, as U is contained in the centre of M. It follows that 〈u,w〉∩M′ is a normal subgroup of M, and hence either 〈u,w〉∩M′={1} or M′≤〈u,w〉. In both cases we get the contradiction that M′ is cyclic. Therefore the statement is proved when the group G is locally finite.
Suppose finally that the group G is locally graded. As the metanorm M has finite commutator subgroup, it cannot be locally nilpotent and so contains a finitely generated non-nilpotent subgroup X. Let a be an arbitrary element of G, and put E=〈X,a〉. Then X is contained in the metanorm M0=M(E) of E, and so it follows from Lemma 4.5 that M′=X′=M0′. Since E/M0 is cyclic, the commutator subgroup E′ is polycyclic-by-finite by Lemma 2.5, and hence E itself is a polycyclic-by-finite group. Thus it is well known that the non-nilpotent subgroup M0 contains a normal subgroup J of E such that M0/J is a finite non-nilpotent group. Since each normal subgroup of E can be obtained as an intersection of normal subgroups of finite index, there exists a normal subgroup D of E such that E/D is finite and M0∩D=J. Let 𝒩 be the set consisting of all E-invariant subgroups of finite index N of D such that M′∩N={1}. As M′ is finite and E is residually finite, the set 𝒩 is not empty and
⋂N∈𝒩N={1}.
Let N be any element of 𝒩. Then M0/J is a homomorphic image of M0N/N, so M0N/N is a non-nilpotent subgroup of the metanorm M1/N of E/N, and again Lemma 4.5 yields that
(M1/N)′=M0′N/N=M′N/N.
Since E/N is finite, it follows from the first part of the proof that M1/M′N is a central section of E, so in particular [M,E] is contained in M′N. Therefore
[M,a]≤[M,E]≤⋂N∈𝒩M′N=M′,
so a acts trivially on M/M′ and hence M/M′ is contained in the centre of G/M′.
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Corollary 4.11.
Let G be a locally graded group whose metanorm M=M(G) is not nilpotent. Then the subgroup ζ2(M) is contained in ζ2(G).
Proof.
Since M/M′ lies in Z(G/M′) by Theorem 4.10, the subgroup [ζ2(M),G] is contained in ζ2(M)∩M′. On the other hand, M′/M′′ is an eccentric chief factor of M, so the intersection ζ2(M)∩M′ is contained in M′′, and so also in Z(G) by Theorem 4.2. Therefore [ζ2(M),G,G]=1 and hence ζ2(M) is a subgroup of ζ2(G).
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