Abstract
A group G is almost Engel if for every \(g\in G\) there is a finite set \({\mathcal {E}}(g)\) such that for every \(x\in G\) all sufficiently long commutators \([x,{}_n\, g]\) belong to \({\mathcal {E}}(g)\), that is, for every \(x\in G\) there is a positive integer n(x, g) such that \([x,{}_n\, g]\in {\mathcal {E}}(g)\) whenever \(n(x,g)\le n\). A group G is almost nil if it is almost Engel and for every \(g\in G\) there is a positive integer n depending on g such that \([x,{}_s g]\in {\mathcal {E}}(g)\) for every \(x\in G\) and every \(s\ge n\). We prove that if a linear group G is almost Engel, then G is finite-by-hypercentral. If G is almost nil, then G is finite-by-nilpotent.
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1 Introduction
By a linear group we understand here a subgroup of GL(m, F) for some field F and a positive integer m. An element g of a group G is called a (left) Engel element if for any \(x\in G\) there exists \(n=n(x,g)\ge 1\) such that \([x,{}_n\, g]=1\). As usual, the commutator \([x,{}_n\, g]\) is defined recursively by the rule
assuming \([x,{}_0\, g]=x\). If n can be chosen independently of x, then g is a (left) n-Engel element. A group G is called Engel if all elements of G are Engel. It is called n-Engel if all its elements are n-Engel. A group is said to be locally nilpotent if every finite subset generates a nilpotent subgroup. Clearly, any locally nilpotent group is an Engel group. It is a long-standing problem whether any n-Engel group is locally nilpotent. Engel linear groups are known to be locally nilpotent (cf. [2, 3]).
We say that a group G is almost Engel if for every \(g\in G\) there is a finite set \({\mathcal {E}}(g)\) such that for every \(x\in G\) all sufficiently long commutators \([x,{}_n\, g]\) belong to \({\mathcal {E}}(g)\), that is, for every \(x\in G\) there is a positive integer n(x, g) such that \([x,{}_n\, g]\in {\mathcal {E}}(g)\) whenever \(n(x,g)\le n\). (Thus, Engel groups are precisely the almost Engel groups for which we can choose \({\mathcal {E}}(g)=\{ 1\}\) for all \(g\in G\).) We say that a group G is nil if for every \(g\in G\) there is a positive integer n depending on g such that g is n-Engel. The group G will be called almost nil if it is almost Engel and for every \(g\in G\) there is a positive integer n depending on g such that \([x,{}_s\,g]\in {\mathcal {E}}(g)\) for every \(x\in G\) and every \(s\ge n\).
Almost Engel groups were introduced in [6] where it was proved that an almost Engel compact group is necessarily finite-by-(locally nilpotent). The purpose of the present article is to prove the following related result.
Theorem 1.1
Let G be a linear group.
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1.
If G is almost Engel, then G is finite-by-hypercentral.
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2.
If G is almost nil, then G is finite-by-nilpotent.
Recall that the union of all terms of the (transfinite) upper central series of G is called the hypercenter. The group G is hypercentral if it coincides with its hypercenter. The hypercentral groups are known to be locally nilpotent (see [10, P. 365]). By well-known results obtained in [2, 3], if under the hypotheses of Theorem 1.1 the group G is Engel or nil, then G is hypercentral or nilpotent, respectively.
As a warning to the reader, we mention that in many articles (including some of the author) the expression “the group G is almost an X-group” for a property X means “G has an X-subgroup of finite index”. In the present paper, however, the meaning of the term “almost Engel” is different. It is hoped that this discrepancy does not lead to a confusion.
2 Preliminaries
Let G be a group and \(g\in G\) an almost Engel element, so that there is a finite set \({\mathcal {E}}(g)\) such that for every \(x\in G\) there is a positive integer n(x, g) with the property that \([x,{}_n\, g]\) belongs to \({\mathcal {E}}(g)\) whenever \(n(x,g)\le n\). If \({\mathcal {E}}'(g)\) is another finite set with the same property for possibly different numbers \(n'(x,g)\), then \({\mathcal {E}}(g)\cap {\mathcal {E}}'(g)\) also satisfies the same condition with the numbers \(n''(x,g)=\max \{n(x,g),n'(x,g)\}\). Hence there is a minimal set with the above property. The minimal set will again be denoted by \({\mathcal {E}}(g)\) and, following [6], called the Engel sink for g, or simply g -sink for short. From now on we will always use the notation \({\mathcal {E}}(g)\) to denote the (minimal) Engel sinks. In particular, it follows that for each \(x\in {\mathcal {E}}(g)\) there exists \(y\in {\mathcal {E}}(g)\) such that \(x=[y,g]\). More generally, given a subset \(K\subseteq G\) and an almost Engel element \(g\in G\), we write \({\mathcal {E}}(g,K)\) to denote the minimal subset of G with the property that for every \(x\in K\) there is a positive integer n(x, g) such that \([x,{}_n\, g]\) belongs to \({\mathcal {E}}(g,K)\) whenever \(n(x,g)\le n\). Throughout the article we use the symbols \(\langle X\rangle \) and \(\langle X^G\rangle \) to denote the subgroup generated by a set X and the minimal normal subgroup of G containing X, respectively.
A group is said to have a property virtually if some subgroup of finite index has the property. The following lemma can be found in [8, Ch. 12, Lemma 1.2] or in [5, Lemma 21.1.4].
Lemma 2.1
A virtually abelian group contains a characteristic abelian subgroup of finite index.
As usual, we write \(Z_i(G)\) for the ith term of the upper central series of G and \(\gamma _i(G)\) for the ith term of the lower central series. A well-known theorem of Schur states that if G is central-by-finite, then the commutator subgroup \(G'\) is finite (see [10, 10.1.4]). Baer proved that if, for a positive integer k, the quotient \(G/Z_k(G)\) is finite, then so is \(\gamma _{k+1}(G)\) (see [10, 14.5.1]). Recently, the following related result was obtained in [1] (see also [7]).
Theorem 2.2
Let G be a group and let H be the hypercenter of G. If G / H is finite, then G has a finite normal subgroup N such that G / N is hypercentral.
We will also require the Dicman Lemma (see [10, 14.5.7]).
Lemma 2.3
In any group a normal finite subset consisting of elements of finite order generates a finite subgroup.
In [9] Plotkin proved that if a group G has an ascending series whose quotients locally satisfy the maximal condition, then the Engel elements of G form a locally nilpotent subgroup. In particular we have the following lemma.
Lemma 2.4
Let G be a group having an ascending series whose quotients locally satisfy the maximal condition and let \(a\in G\) be an Engel element. Then \(\langle a^G\rangle \) is locally nilpotent.
Linear groups are naturally equipped with the Zariski topology. If G is a linear group, the connected component of G containing 1 is denoted by \(G^0\). We will use (sometimes implicitly) the following facts on linear groups. All these facts are well-known and are provided here just for the reader’s convenience.
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If G is a linear group and N a normal subgroup which is closed in the Zarissky topology, then G / N is linear (see [12, Theorem 6.4]).
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Since finite subsets of G are closed in the Zariski topology, it follows that any finite subgroup of a linear group is closed. Hence G / N is linear for any finite normal subgroup N.
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If G is a linear group, the connected component \(G^0\) has finite index in G (see [12, Lemma 5.3]).
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Each finite conjugacy class in a linear group centralizes \(G^0\) (see [12, Lemma 5.5]).
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In a linear group any descending chain of centralizers is finite.
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A linear group generated by normal nilpotent subgroups is nilpotent (see Gruenberg [3]).
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Tits alternative: A finitely generated linear group either is virtually soluble or contains a subgroup isomorphic to a nonabelian free group (see [11]).
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The Burnside–Schur theorem: A periodic linear group is locally finite (see [12, 9.1]).
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Zassenhaus theorem: A locally soluble linear group is soluble. Every linear group contains a unique maximal soluble normal subgroup (see [12, Corollary 3.8]).
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Since the closure in the Zariski topology of a soluble subgroup is again soluble (see [12, Lemma 5.11]), it follows that the unique maximal soluble normal subgroup of a linear group is closed. In particular, if G is linear and R is the unique maximal soluble normal subgroup of G, then G / R is linear and has no nontrivial normal soluble subgroups.
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A locally nilpotent linear group is hypercentral (see [2] or [3]).
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Gruenberg theorem: The set of Engel elements in a linear group G coincides with the Hirsch–Plotkin radical of G. The set of right Engel elements coincides with the hypercenter of G (see [3]).
Here, as usual, the Hirsch–Plotkin radical of a group is the maximal normal locally nilpotent subgroup. An element \(g\in G\) is a right Engel element if for each \(x\in G\) there exists a positive integer n such that \([g,{}_n\, x]=1\).
3 Almost Engel elements in virtually soluble groups
In the present section we give certain criteria for a group containing almost Engel elements to be finite-by-nilpotent or finite-by-hypercentral. In particular, we prove that a virtually soluble group generated by finitely many almost Engel elements is finite-by-nilpotent (Theorem 3.3).
Lemma 3.1
Let \(G=H\langle a_1,\dots ,a_s\rangle \), where H is a normal subgroup and \(a_i\) are almost Engel elements. Assume that G / H is nilpotent. If \(N\le H\) is a finite normal subgroup of H, then \(\langle N^G\rangle \) is finite.
Proof
Suppose first that \(s=1\) and write a in place of \(a_1\). Let M be the subgroup generated by all commutators of the form \([x,{}_j\, a]\), where \(x\in N\) and j is a nonnegative integer. Since both N and \({\mathcal {E}}(a)\) are finite, it follows that there exists an integer k such that M is contained in the product \(\prod _{i=0}^kN^{a^i}\). It is clear that the product \(\prod _{i=0}^kN^{a^i}\) is normal in H and a normalizes M. Therefore \(\langle M^H\rangle \) is normal in G and is contained in \(\prod _{i=0}^kN^{a^i}\). Moreover, \(\langle N^G\rangle =\langle M^H\rangle \) so in the case where \(s=1\) the lemma follows.
Therefore we will assume that \(s\ge 2\) and use induction on s. Assume additionally that G / H is abelian. Set \(H_0=H\) and \(H_i=H_{i-1}\langle a_i\rangle \) for \(i=1,\dots ,s\). The subgroups \(H_i\) are normal in G and \(H_s=G\). By induction, \(K=\langle N^{H_{s-1}}\rangle \) is finite. Since \(G=H_{s-1}\langle a_s\rangle \), the above paragraph shows that \(\langle K^G\rangle \) is finite. Obviously, \(\langle K^G\rangle =\langle N^G\rangle \) and so in the case where G / H is abelian the lemma follows.
We will now allow G / H to be nonabelian, say of nilpotency class c. We will use induction on c. Set \(B=\langle a_s^G\rangle \) and \(G_1=HB\). Since G / H is a finitely generated nilpotent group, it follows that each subgroup of G / H is finitely generated and so B has finitely many conjugates of \(a_s\), say \(a_s^{g_1}\dots ,a_s^{g_r}\) such that \(G_1=H\langle a_s^{g_1}\dots ,a_s^{g_r}\rangle \). Since \(G_1/H\) has nilpotency class at most \(c-1\), by induction \(\langle N^{G_1}\rangle \) is finite. We now note that \(G=G_1\langle a_1,\dots ,a_{s-1}\rangle \) so the induction on s completes the proof. \(\square \)
Lemma 3.2
Let \(G=H\langle a\rangle \), where H is a virtually abelian normal subgroup and a is an almost Engel element. Then \(\langle a^G\rangle \) is finite-by-(locally nilpotent).
Proof
Assume that G is a counter-example with \(|{\mathcal {E}}(a)|\) as small as possible. In view of Lemma 2.1 we can choose a maximal characteristic abelian subgroup V in H. Since V is abelian, we have \([v_1,a][v_2,a]=[v_1v_2,a]\) for any \(v_1,v_2\in V\). In other words, a product of two commutators of the form [v, a], where \(v\in V\), again has the same form. Therefore \({\mathcal {E}}(a,V)\) is a finite subgroup. Obviously, the normalizer in G of \({\mathcal {E}}(a,V)\) has finite index. It follows that \({\mathcal {E}}(a,V)\) is contained in a finite normal subgroup N. If \({\mathcal {E}}(a,V)\ne 1\), we pass to the quotient G / N and use induction on \(|{\mathcal {E}}(a)|\). Therefore without loss of generality we will assume that \({\mathcal {E}}(a,V)=1\), that is, a is Engel in \(V\langle a\rangle \). Since \({\mathcal {E}}(a)\) consists of commutators of the form [x, a] with \(x\in {\mathcal {E}}(a)\), it follows that \({\mathcal {E}}(a)\cap V=\{1\}\). Let \(C_0=1\) and
for \(i=1,2,\dots \). Since a is Engel in V, we have \(V=\cup _i C_i\).
Let \(T=\langle {\mathcal {E}}(a),a\rangle \) and \(U=V\cap T\). We observe that U is a finitely generated abelian subgroup. In view of the fact that V is the union of the \(C_i\) we deduce that there exists a positive integer n such that \(U=C_n\cap U\).
For \(i=0,\dots ,n\) set \(U_i=C_i\cap U\). Thus, \(U=U_n\). Observe that \(U_1\) centralizes a and therefore \(U_1\) normalizes the set \({\mathcal {E}}(a)\). Denote by \(W_{1}\) the intersection \(U_1\cap C_G({\mathcal {E}}(a))\). Since \({\mathcal {E}}(a)\) is finite, it follows that \(W_1\) has finite index in \(U_1\). Further, it is clear that \(W_1\) is contained in the center Z(T).
The finiteness of the index \([U_1:W_1]\) implies that \(U_2\) contains a normal in T subgroup \(W_2\) such that the index \([U_2:W_2]\) is finite, and \([W_2,T]\le W_1\). Thus, \(W_2\) is contained in \(Z_2(T)\), the second term of the upper central series of T.
Next, in a similar way we conclude that \(U_3\cap Z_3(T)\) has finite index in \(U_3\) and so on. Eventually, we deduce that \(U\cap Z_n(T)\) has finite index in U. Thus, \(T/Z_n(T)\) is finite-by-cyclic and therefore there exists a positive integer k such that \(a^k\in Z_{n+1}(T)\). Hence, \(T/Z_{n+1}(T)\) is finite and so, in view of Baer’s theorem, we deduce that T is finite-by-nilpotent. In particular, for some positive integer r the subgroup \(\gamma _r(T)\) is finite. The observation that for each \(x\in {\mathcal {E}}(a)\) there exists \(y\in {\mathcal {E}}(a)\) such that \(x=[y,g]\) guarantees that \(\mathcal E(a)\) is contained in \(\gamma _r(T)\). In particular, we proved that the subgroup \(\langle {\mathcal {E}}(a)\rangle \) is finite. Because V is abelian, it is obvious that V normalizes \(V\cap \langle {\mathcal {E}}(a)\rangle \). Thus, \(V\cap \langle \mathcal E(a)\rangle \) is a finite subgroup with normalizer of finite index. It follows that \(V\cap \langle \mathcal E(a)\rangle \) is contained in a finite normal subgroup of G. We can factor out the latter and without loss of generality assume that \(V\cap \langle {\mathcal {E}}(a)\rangle =1\).
Recall that \(C_1=C_V(a)\). Therefore \(C_1\) normalizes \(\langle {\mathcal {E}}(a)\rangle \) and in view of the fact that \(V\cap \langle {\mathcal {E}}(a)\rangle =1\) we conclude that \(C_1\) centralizes \(\langle {\mathcal {E}}(a)\rangle \). So \(C_1\le Z(VT)\). Same argument shows that \(C_2/C_1\le Z(VT/C_1)\) and, more generally, \(C_{i+1}/C_i\le Z(VT/C_i)\) for \(i=0,1,2\dots \). Thus, \(V\le Z_\infty (VT)\) where \(Z_\infty (VT)\) stands for the hypercenter of T. Of course, it follows that there exists a positive integer k such that \(a^k\in Z_\infty (VT)\). We deduce that \(Z_\infty (VT)\) has finite index in VT. Theorem 2.2 now tells us that VT has a finite normal subgroup N such that the quotient group (VT) / N is hypercentral. The hypercentral groups are locally nilpotent and so VT is finite-by-(locally nilpotent). The observation that for each \(x\in {\mathcal {E}}(a)\) there exists \(y\in {\mathcal {E}}(a)\) such that \(x=[y,g]\) guarantees that \({\mathcal {E}}(a)\) is contained in N.
Since VT has finite index in G, Dicman’s lemma tells us that G contains a finite normal subgroup R such that \({\mathcal {E}}(a)\subseteq N\le R\). The image of a in G / R is Engel and the required result follows from Lemma 2.4. \(\square \)
Theorem 3.3
A virtually soluble group generated by finitely many almost Engel elements is finite-by-nilpotent.
Proof
Let G be a virtually soluble group generated by finitely many almost Engel elements \(a_1,\dots ,a_s\) and let S be a normal soluble subgroup of finite index in G. We assume that \(S\ne 1\) and let V be the last nontrivial term of the derived series of S. By induction on the derived length of S we assume that G / V is finite-by-nilpotent. Therefore G contains a normal subgroup H such that V has finite index in H and the quotient G / H is nilpotent. For \(i=1,\dots ,s\) set \(G_i=H\langle a_i\rangle \). By Lemma 3.2 each subgroup \(\langle a_i^{G_i}\rangle \) has a finite normal subgroup \(N_i\) such that \(\langle a_i^{G_i}\rangle /N_i\) is locally nilpotent. Since \(G_i/H\) are abelian, it is clear that all quotients \(G_i/H\cap N_i\) are locally nilpotent and so, replacing if necessary \(N_i\) by \(H\cap N_i\), without loss of generality we can assume that all subgroups \(N_i\) are normal subgroups of H. Therefore the product of the subgroups \(N_i\) is finite. By Lemma 3.1 the product of \(N_1\cdots N_s\) is contained in a finite subgroup N which is normal in G. Obviously the images in G / N of the generators \(a_1,\dots ,a_s\) are Engel. Thus, G / N is a virtually soluble group generated by finitely many Engel elements. It follows from Lemma 2.4 that G / N is nilpotent. The proof is complete. \(\square \)
The next lemma is well-known. For the reader’s convenience we provide the proof.
Lemma 3.4
Let \(G=H\langle a\rangle \), where H is a nilpotent normal subgroup and a is a nil element. Then G is nilpotent.
Proof
Suppose that a is n-Engel. Let \(K=Z(H)\) and set \(K_0=K\) and \(K_{i+1}=[K_i,a]\) for \(i=0,1,\dots \). Then \(K_{n-1}\le K\cap C_K(a)\) and so \(K_{n-1}\le Z(G)\). Moreover we observe that \([K_{i-1},G]\le K_i\) and it follows that \(K_{n-i}\le Z_{i}(G)\) for \(i=1,2,\dots ,n\). Therefore \(K\le Z_{n}(G)\). Passing to the quotient \(G/Z_{n}(G)\) and using induction on the nilpotency class of H we deduce that if H is nilpotent with class c, then G is nilpotent with class at most cn. \(\square \)
Lemma 3.5
Let \(G=H\langle a\rangle \), where H is a hypercentral normal subgroup and a is an Engel element. Then G is hypercentral.
Proof
It is sufficient to show that \(Z(G)\ne 1\). Let \(Z=Z(H)\). Since a is an Engel element, \(C_Z(a)\ne 1\). Obviously, \(C_Z(a)\le Z(G)\). The proof is complete. \(\square \)
Lemma 3.6
Let a be an almost Engel element in a group G and assume that \({\mathcal {E}}(a)\) is contained in a locally nilpotent subgroup. Then the subgroup \(\langle {\mathcal {E}}(a)\rangle \) is finite.
Proof
Set \(D=\langle {\mathcal {E}}(a)\rangle \). Without loss of generality we can assume that \(G=D\langle a\rangle \). Since \({\mathcal {E}}(a)\) is finite, D is nilpotent and we can use induction on the nilpotency class of D. Thus, by induction assume that the quotient of D over its center is finite. By Schur’s theorem the derived group \(D'\) is finite as well. Factoring out \(D'\) we can assume that D is abelian. So now D is abelian and \(D=[D,a]\). By [6, Lemma 2.3], \(D={\mathcal {E}}(a)\) and hence D is finite. \(\square \)
Lemma 3.7
Let \(G=H\langle a\rangle \), where H is a hypercentral normal subgroup.
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1.
If a is almost Engel, then G is finite-by-hypercentral.
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2.
If H is nilpotent and a is almost nil, then G is finite-by-nilpotent.
Proof
We will prove Claim 1 first. Assume that a is almost Engel. Let N be the product of all normal subgroups of G whose intersection with \({\mathcal {E}}(a)\) is \(\{1\}\). It is easy to see that \(N\cap \mathcal E(a)=\{1\}\) and N is the unique maximal normal subgroup with that property. Therefore \(K\cap {\mathcal {E}}(a)\ne \{1\}\) whenever K is a normal subgroup containing N as a proper subgroup. Since \({\mathcal {E}}(a)\) is finite, the group G contains a minimal normal subgroup M such that \(N<M\). Taking into account that H is hypercentral, we observe that M / N is central in H / N.
Let \(D=\langle {\mathcal {E}}(a)\rangle \cap M\). It follows that \(M=ND\). Suppose that D is not normal in M and set \(L=N_M(N_M(D))\). Since M is hypercentral, it satisfies the normalizer condition and so \(L\ne N_M(D)\). Obviously a normalizes both L and \(N_M(D)\). Since a acts on \(L/N_M(D)\) as an Engel element, the centralizer of a in \(L/N_M(D)\) is nontrivial. Thus, L has a subgroup C such that \(N_M(D)<C\) and C normalizes \(N_M(D)\langle a\rangle \). Of course, D is normal in \(N_M(D)\langle a\rangle \). By Lemma 3.5 the quotient of \(N_M(D)\langle a\rangle \) by D is hypercentral. It is easy to see that D is a unique minimal normal subgroup of \(N_M(D)\langle a\rangle \) whose quotient is hypercentral. Therefore D is characteristic in \(N_M(D)\langle a\rangle \) and so C normalizes D. This is a contradiction since \(N_M(D)<C\).
Hence, D is normal in M. Again, it is easy to see that D is a unique minimal normal subgroup of \(M\langle a\rangle \) whose quotient is hypercentral. Therefore D is characteristic in M and so it is normal in G. We pass to the quotient G / D and Claim 1 now follows by straightforward induction on \(|{\mathcal {E}}(a)|\).
We now assume that H is nilpotent and a is almost nil. We already know that G is finite-by-hypercentral. Factoring out a finite normal subgroup we can assume that G is hypercentral. In that case a is actually nil and so by Lemma 3.4 G is nilpotent. The proof of the lemma is complete. \(\square \)
4 Linear groups
Lemma 4.1
A virtually soluble almost Engel linear group is finite-by-hypercentral.
Proof
Suppose that G is a virtually soluble almost Engel linear group. Let S be a normal soluble subgroup of finite index in G. By induction on the derived length of S we assume that \(S'\) is finite-by-hypercentral. Passing to the quotient over a normal finite subgroup without loss of generality we can assume that \(S'\) is hypercentral. By Lemma 3.7 the subgroup \(\langle S',x\rangle \) is finite-by-hypercentral for each \(x\in G\). Thus, for each \(x\in G\) there exists a finite characteristic subgroup \(R_x\le \langle S',x\rangle \) such that \(\langle S',x\rangle /R_x\) is hypercentral. Since \(\langle S',x\rangle \) is normal in S, it follows that each element in \(R_x\) has centralizer of finite index in S, hence centralizer of finite index in G. Therefore \(G^0\) centralizes \(R_x\) and it follows that \(\langle S',x\rangle \) is hypercentral for each \(x\in G^0\). The subgroup \(\prod \langle S',x\rangle \), where x ranges over \(S\cap G^0\), is locally nilpotent and therefore hypercentral. In particular \(N=S\cap G^0\) is hypercentral and so G is virtually hypercentral. By Lemma 3.7 the subgroup \(\langle N,x\rangle \) is finite-by-hypercentral for each \(x\in G\). In other words, for each \(x\in G\) there exists a finite characteristic subgroup \(Q_x\le \langle N,x\rangle \) such that the quotient \(\langle N,x\rangle /Q_x\) is hypercentral. Since N has finite index in G, it follows that G contains only finitely many subgroups of the form \(\langle N,x\rangle \). Set \(N_0=\prod _{x\in G}Q_x\). We see that \(N_0\) is a finite normal subgroup. Pass to the quotient \(G/N_0\). Now the subgroup \(\langle N,x\rangle \) is hypercentral for each \(x\in G\). It follows that N consists of right Engel elements and so, by the result of Gruenberg, N is contained in the hypercenter of G. It follows from Theorem 2.2 that G is finite-by-hypercentral, as required. \(\square \)
We are now ready to prove Theorem 1.1 in its full generality. For the reader’s convenience we restate it here.
Theorem 4.2
Let G be a linear group. If G is almost Engel, then G is finite-by-hypercentral. If G is almost nil, then G is finite-by-nilpotent.
Proof
Assume that G is almost Engel. In view of Lemma 4.1 it is sufficient to show that G is virtually soluble. By the Zassenhaus theorem a linear group is soluble if and only if it is locally soluble. Therefore it is sufficient to show that G is virtually locally soluble. It is clear that G does not contain a subgroup isomorphic to a nonabelian free group. Hence, by Tits alternative, any finitely generated subgroup of G is virtually soluble. Therefore, by Theorem 3.3, any finitely generated subgroup of G is finite-by-nilpotent. It becomes obvious that elements of finite order in G generate a periodic subgroup. Moreover, the quotient of G over the subgroup generated by all elements of finite order is locally nilpotent. Hence, G is virtually locally soluble if and only if so is the subgroup generated by elements of finite order. Therefore without loss of generality we can assume that G is an infinite periodic (and locally finite) group.
Let R be the soluble radical of G. We can pass to the quotient and without loss of generality assume that \(R=1\). So in particular G has no nontrivial Engel elements. By the theorem of Hall–Kulatilaka G contains an infinite abelian subgroup [4]. We conclude that some centralizers in G are infinite. Since G satisfies the minimal condition on centralizers, it follows that G has a subgroup \(D\ne 1\) such that the centralizer \(C=C_G(D)\) is infinite while \(C_G(\langle D,x\rangle )\) is finite for each \(x\in G\setminus D\). Using that C is infinite we deduce from the Hall–Kulatilaka theorem that C contains an infinite abelian subgroup A. Obviously \(A\le C_G(\langle D,A\rangle )\) and it follows that \(A\le D\). Thus, \(A\le Z(C)\).
Now choose \(1\ne a\in A\). The centralizer C normalizes the finite set \(\mathcal E(a)\) because \(a\in Z(C)\). Hence, C contains a subgroup of finite index which centralizes \({\mathcal {E}}(a)\). It follows that \(C_G(\langle D,{\mathcal {E}}(a)\rangle )\) is infinite and we conclude that \(\mathcal E(a)\) is contained in D and C centralizes \({\mathcal {E}}(a)\). In particular, a centralizes \({\mathcal {E}}(a)\) and so \({\mathcal {E}}(a)=\{1\}\). Thus, a is an Engel element, a contradiction. This completes the proof of Claim 1.
Suppose now that G is almost nil. We already know that G is finite-by-hypercentral. Passing to a quotient over a finite normal subgroup we can assume that G is hypercentral. Then obviously G, being both hypercentral and almost nil, must be nil. By the result of Gruenberg, G is nilpotent. \(\square \)
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Communicated by A. Constantin.
This research was supported by FAPDF and CNPq-Brazil.
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Shumyatsky, P. Almost Engel linear groups. Monatsh Math 186, 711–719 (2018). https://doi.org/10.1007/s00605-017-1062-x
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DOI: https://doi.org/10.1007/s00605-017-1062-x