Groups in which each subgroup is commensurable with a normal subgroup
aa r X i v : . [ m a t h . G R ] M a y Groups in which each subgroup iscommensurable with a normal subgroup to the memory of Jim Wiegold
Carlo Casolo, Ulderico Dardano, Silvana Rinauro
Abstract
A group G is a cn -group if for each subgroup H of G there existsa normal subgroup N of G such that the index | HN : ( H ∩ N ) | isfinite. The class of cn -groups contains properly the classes of core-finite groups and that of groups in which each subgroup has finiteindex in a normal subgroup.In the present paper it is shown that a cn -group whose periodicimages are locally finite is finite-by-abelian-by-finite. Such groupsare then described into some details by considering automorphisms ofabelian groups. Finally, it is shown that if G is a locally graded groupwith the property that the above index is bounded independently of H , then G is finite-by-abelian-by-finite. Key words and phrases : locally finite, core-finite, subnormal, inert, cf -group. : Primary 20F24, Secondary20F18, 20F50, 20E15 In a celebrated paper, B.H.Neumann [10] showed that for a group G theproperty that each subgroup H has finite index in a normal subgroup of G (i.e., | H G : H | is finite) is equivalent to the fact that G has finite derivedsubgroup ( G is finite-by-abelian ).A class of groups with a dual property was considered in [1]. A group G is said to be a cf -group ( core-finite ) if each subgroup H contains a normalsubgroup of G with finite index in H (i.e., | H : H G | is finite). As Tarskigroups are cf , a complete classification of cf -groups seems to be much dif-ficult. However, in [1] and [12] it has been proved that a cf -group G whose eriodic quotients are locally finite is abelian-by-finite and, if G is periodic,there exists an integer n such that | H : H G | ≤ n for all H ≤ G (say that G is bcf , boundedly cf ) and that a locally graded bcf -group is abelian-by-finite .Furthermore, an easy example of a metabelian (and even hypercentral) groupwhich is cf but not bcf is given. It seems to be a still open question whetherevery locally graded cf -group is abelian-by-finite. Recall that a group is saidto be abelian-by-finite if has an abelian subgroup with finite index and thata group is said to be locally finite ( locally graded , respectively) if each non-trivial finitely generated subgroup is finite (has a proper subgroup with finiteindex, respectively).With the aim of considering the above properties in a common framework,recall that two subgroups H and K of a group G are said to be commensurable if H ∩ K has finite index in both H and K . This is an equivalence relationand will be denoted by ∼ . Clearly, if H ∼ K , then ( H ∩ L ) ∼ ( K ∩ L ) and HM ∼ KM for each L ≤ G and M ⊳ G .Thus, in the present paper we consider the class of cn -groups , that is, groups in which each subgroup is commensurable with a normal subgroup .Into details, for a subgroup H of a group G define δ G ( H ) to be the minimumindex | HN : ( H ∩ N ) | with N ⊳ G . Then G is a cn -group if and only if δ G ( H ) is finite for all H ≤ G . Clearly, subgroups and quotients of cn -groupsare also cn -groups.Note that if a subgroup H of a group G is commensurable with a normalsubgroup N , then S := ( H ∩ N ) N has finite index in H . Thus the class of cn -groups is contained in the class of sbyf-groups , that is, groups in whicheach subgroup H contains a subnormal subgroup S of G such that the index | H : S | is finite (i.e., H is subnormal-by-finite ). It is known that locallyfinite sbyf-groups are (locally nilpotent)-by-finite (see [7]) and nilpotent-by-Chernikov (see [3]).The extension of a finite group by a cn -group is easily seen to be a cn -group, see Proposition 1.1 below. Moreover, from Proposition 9 in [4] it fol-lows that for an abelian-by-finite group properties cn and cf are equivalent .However, for each prime p there is a nilpotent p -group with the property cn which is neither finite-by-abelian nor abelian-by-finite, see Proposition 1.2.Our main result is the following. Theorem A
Let G be a cn -group such that every periodic image of G islocally finite. Then G is finite-by-abelian-by-finite. Here by a finite-by-abelian-by-finite group we mean a group which has a2nite-by-abelian subgroup of finite index. The proof of Theorem A will begiven in Sect. 3. The strategy of the proof will be to reduce to the case when G is nilpotent and then to apply techniques of nilpotent groups theory. Tothis end, in Sect. 2, we will study the action of a cn -group on its abeliansections.We will consider also bcn -groups , that is, groups G for which there is n ∈ N such that δ G ( H ) ≤ n for all H ≤ G and prove the following theorem. Theorem B
Let G be a finite-by-abelian-by-finite group.i) G is cn if and only if it is finite-by- cf .ii) G is bcn if and only if it is finite-by- bcf . It follows that if the group G is periodic and finite-by-abelian-by-finite,then G is bcn if and only if it is cn . Then we consider non-periodic finite-by-abelian-by-finite bcf -groups in Proposition 3.2.The more restrictive property bcn remains treatable when we considerthe wider class of locally graded groups. Theorem C
A locally graded bcn -group is finite-by-abelian-by-finite.
Preliminaries
Our notation is mostly standard. For undefined terminology and basicfacts we refer to [11]. If Γ is a group acting on a group G and H ≤ G , wedenote H Γ := ∩ γ ∈ Γ H γ and H Γ := h H γ | γ ∈ Γ i . We say that H is Γ -invariant (or a Γ -subgroup ) if H Γ = H .We first point out a sufficient condition for a group to be cn (or even bcn ) and give examples of non trivial cn -groups. Proposition 1.1
Let G be a group with a normal series G ≤ G ≤ G ,where G and G/G have finite order, m and n respectively.If H ≤ G , then H is commensurable with H := ( H ∩ G ) G ≤ G and δ G ( H ) ≤ mn · δ G/G ( H /G ) .In particular, if each subgroup of G /G is commensurable with a normalsubgroup of G/G , then G is a cn -group. (cid:3) Proposition 1.2
For each prime p there is a nilpotent p -group with theproperty bcn , which is neither abelian-by-finite nor finite-by-abelian. roof. Consider a sequence P n of isomorphic groups with order p definedby P n := h x n , y n | x p n = y pn = 1 , x y n n = x p n i = h x n i ⋊ h y n i where clearly P ′ n = h x p n i has order p . Let P := Dr n ∈ N P n and consider the automorphism γ of P such that x γn = x pn and y γn = y n , for each n ∈ N . Clearly, γ hasorder p , acts as the automorphism x x p on P/P ′ (which has exponent p ) and acts trivially on P ′ (which is elementary abelian). Finally let N := h x p x p n | n ∈ N i . Then N is a γ -invariant subgroup of P ′ with index p . Thusthe p -group G := ( P ⋊ h γ i ) /N is a bcn -group by Proposition 1.1 applied tothe series P ′ /N ≤ P/N ≤ G .We have that G ′ is infinite, since for each n we have x pn = [ x n , γ ] ∈ [ P n , γ ] > P ′ n . Moreover, we have that gN ∈ Z ( P/N ) if and only if ∀ i [ g, P i ] ≤ N , and N ∩ P i = 1. Thus Z ( P/N ) = Z ( P ) /N where Z ( P ) = Dr n h x pn i hasinfinite index in P .If, by contradiction, G is abelian-by-finite, then there is an abelian normalsubgroup A/N of P/N with finite index. Then for some m ∈ N we have P = AF , where F = Dr n Recall that an automorphism γ of a group A is said to be a power auto-morphism if H γ = H for each subgroup H ≤ A . It is well-known (see [11])that, if A is an abelian p -group, then there exists a p -adic integer α such that a γ = a α for all a ∈ A . Here a α stands for a n , where n is any integer congruentto α modulo the order of a . On the other hand, a power automorphism of anon-periodic abelian group is either the identity or the inversion map.As in [4], if Γ is a group acting on an abelian group A , we consider thefollowing properties: p ) ∀ H ≤ A H = H Γ ; ap ) ∀ H ≤ A | H : H Γ | < ∞ ; bp ) ∀ H ≤ A | H Γ : H | < ∞ ; cp ) ∀ H ≤ A ∃ K = K Γ ≤ A such that H ∼ K , ( H , K are commensurable ).Obviously both ap and bp imply cp . Moreover, from Propositions 8and 9 in [4] it follows that these three properties are equivalent, provided A is abelian and Γ is finitely generated, while they are in fact different in thegeneral case even when A and Γ are elementary abelian p -groups . On theother hand, the properties ap and bp have been previously characterized in46] and [2] respectively, as we are going to recall.To shorten statements we define a further property:˜ p ) Γ has p on the factors of a Γ -series ≤ V ≤ D ≤ A wherei) V is free abelian of finite rank,ii) D/V is divisible periodic with finite total rank,iii) A/D is periodic and has finite p -exponent for each prime p ∈ π ( D/V ) . Theorem 2.1 [6],[2] Let Γ be group acting on an abelian group A . Then:a) Γ has ap on A if and only if there is a Γ -subgroup A such that A/A isfinite and Γ has either p or ˜ p on A .b) Γ has bp on A if and only if there is a Γ -subgroup A such that A isfinite and Γ has either p or ˜ p on A/A . By next statement we give a characteration of the property cp along thesame lines. Theorem 2.2 Let Γ be group acting on an abelian group A . Then:c) Γ has cp on A if and only if there are Γ -subgroups A ≤ A ≤ A suchthat A and A/A are finite and Γ has either p or ˜ p on A /A . The proof of Theorem 2.2 is at the end of this section. Here we deduce acorollary. Corollary 2.3 For a group Γ acting on an abelian group A , the followingare equivalent:a) Γ has ap on A/A for a finite Γ -subgroup A of A ,b) Γ has bp on a finite index Γ -subgroup A of A ,c) Γ has cp on A . (cid:3) Let us state a couple of elementary basic facts. Proposition 2.4 Let Γ be group acting on a locally nilpotent periodic group A . Then Γ has ap , bp , cp on A , respectively, if and only if Γ has ap , bp , cp on finitely many primary components of A , respectively, and p on all theother ones. Proof. This proof uses the same argument as in Proposition 4.1 in [5].The sufficiency of the condition is clear once one notes that for each H ≤ A it results H = Dr p ( H ∩ A p ), where A p denotes the p -component of A .5oncerning necessity, suppose Γ does not have p on the primary p -component A p of A for infinitely many primes p . Then for each such p there is H p ≤ A p which is not Γ-invariant. We have that the subgroup generated by the H p ’sis not commensurable to any Γ-subgroup. (cid:3) Lemma 2.5 Let Γ be a group acting on an abelian group A . If Γ has cp on A , then:i) Γ has p on the largest periodic divisible subgroup of A .ii) if A is torsion-free, then each γ ∈ Γ acts on A by either the identity orthe inversion map. Proof. Statement ( i ) follows from Lemma 4.3 in [5]. Concerning ( ii ), byPropositions 3.3 and 3.2 of [5] we have that there are coprime non-zero in-tegers n, m such that a m = ( a γ ) n for each a ∈ A . Consider H such that1 = H := h a i ≤ A . Then there is a Γ-invariant subgroup K of A which iscommensurable with H . Thus there is r ∈ N such that K r is a Γ-invariantnontrivial subgroup of H . This forces mn = ± (cid:3) Now we prove some lemmas. In the first one we do not require that thegroup A is abelian. Lemma 2.6 Let Γ be a group acting on a fc -group A . If Γ has cp on A ,then Γ has bp on the subgroup X := { a ∈ A | h a i Γ is finite } of A . Proof. Notice that X is the set of elements a of finite order of A such that | Γ : C Γ ( a ) | is finite, so X is a locally finite Γ-subgroup of A . For any H ≤ X there is K ≤ X such that H ∼ K = K Γ ≤ A . Then there is a finite subgroup F ≤ X such that H ≤ KF . Thus H Γ ≤ KF Γ and | H Γ : H | ≤ | F Γ |·| HK : H | is finite. (cid:3) Lemma 2.7 Let Γ be a group acting on a p -group A which is the directproduct of cyclic groups. If Γ has cp on A , then the subgroup X := { a ∈ A | h a i Γ is finite } has finite index in A . Proof. Assume by contradiction that A/X is infinite.Let us see, by elementary facts, that there is a sequence ( a n ) of elementsof A such that1) h a n | n ∈ N i = Dr n ∈ N h a n i ,2) A I /A I ∩ X is infinite, for each infinite subset I of N , where A I := h a n | n ∈ I i .6n fact, if A/X has finite rank, it has a Pr¨ufer subgroup Q/X . Let Y bea countable subgroup of A such that Q = Y X . By Kulikov Theorem (see[11]) Y is the direct product of cyclic groups, so that we may choose elements a n ∈ Y such that h a n | n ∈ N i = Dr n ∈ N h a n i ≤ Y and | a n X | < | a n +1 X | . Theclaim holds. Similarly, if A/X has infinite rank, consider a countably infinitesubgroup Q/X of the socle of A/X . As above, let Y be a countable subgroupof A such that Q = Y X . Then we may choose elements a n ∈ Y which areindependent mod X and generate their direct product as claimed.We claim now that there are sequences of infinite subsets I n , J n of N and Γ -subgroups K n ≤ A such that for each n ∈ N : I n ∩ J n = ∅ and I n +1 ⊆ J n K n ∼ A I n 5) ( K . . . K i ) ∩ ( A I . . . A I n ) ≤ ( A I . . . A I i ) , ∀ i ≤ n . To prove the claim, proceed by induction on n . Choose an infinite subset I of N such that J := N \ I is infinite. By cp -property there exists K = K Γ1 commensurable with A I .Suppose we have defined I j , J j and K j for 1 ≤ j ≤ n such that 3-5 hold.Since ( K . . . K n ) ∼ ( A I . . . A I n ), there is m ∈ N such that6) ( K K . . . K n ) ∩ A N ≤ ( A I A I . . . A I n ) h a , . . . , a m i .Let I n +1 and J n +1 be disjoint infinite subsets of J n \ { , . . . , m } . By cp -property there exists K n +1 = K Γ n +1 commensurable with A I n +1 . By the choiceof I n +1 it follows that7) ( K . . . K i ) ∩ ( A I . . . A I n +1 ) ≤ ( K . . . K i ) ∩ ( A I . . . A I n ) ∀ i ≤ n and so (5) holds for n + 1, as whished. The claim is now proved.Note that by (2) and (5) it follows that A I n /A I n ∩ X is infinite for each n ∈ N and that also the following property holds8) ( K K . . . K n ) ∩ ¯ A ≤ ( A I A I . . . A I n ) ∀ n , where ¯ A := Dr n ∈ N A I n .Now for each n ∈ N , choose an element b n ∈ ( A I n ∩ K n ) \ X . Then we have B := h b n | n ∈ N i = Dr n h b n i , where h b n i Γ is infinite and h b n i Γ ≤ K n ∼ A I n ,so that9) h b n i Γ ∩ A I n is infinite for each n .Since there exists B = B Γ0 ∼ B , we may take- B ∗ := ( B ∩ B ) Γ = ( B ∗ ∩ B ) Γ ≤ B Γ where B ∗ ∼ B .Now B ∗ / ( B ∗ ∩ B ) and B/ ( B ∗ ∩ B ) are both finite and there is n ∈ N suchthat if B n := h b , . . . , b n i we have 7 ( B ∗ ∩ B ) Γ = B ∗ ≤ ( B ∗ ∩ B ) B Γ n and- B = ( B ∗ ∩ B ) B n .Since b n ∈ K n for each n , we have B n ≤ ¯ K n := K K . . . K n and- B Γ = ( B ∗ ∩ B ) Γ B Γ n ≤ ( B ∗ ∩ B ) B Γ n ≤ ( B ∗ ∩ B ) ¯ K n ≤ B ¯ K n , so that- B Γ ∩ ¯ A ≤ B ¯ K n ∩ ¯ A = B ( ¯ K n ∩ ¯ A ) ≤ BA I A I . . . A I n by (8) above.Thus- h b n +1 i Γ ∩ A I n +1 ≤ B Γ ∩ A I n +1 ≤ ( BA I A I . . . A I n ) ∩ A I n +1 = h b n +1 i is finite,a contradiction with (9). (cid:3) Lemma 2.8 Let Γ be a group acting on an abelian periodic reduced group A . If Γ has cp on A , then there are Γ -subgroups A ≤ A ≤ A such that A and A/A are finite and Γ has p on A /A . Proof. By Proposition 2.4 it is enough to consider the case when A is a p -group. If A is the direct product of cyclic groups, by Lemma 2.7 we havethat A := { a ∈ A | h a i Γ is finite } has finite index in A . Further, by Lemma2.6, Γ has bp on A . Then the statement follows from Theorem 2.1.Let A be any reduced p -group and B ∗ be a basic subgroup of A . Thenthere is B = B Γ ∼ B ∗ . Since A/B ∗ is divisible, then the divisible radical of A/B has finite index. Thus we may assume that A/B is divisible. By KulikovTheorem (see [11]), also B is a direct product of cyclic groups, therefore bythe above there are Γ-subgroups B ≤ B ≤ B such that B and B/B arefinite and Γ has p on B /B . We may assume B = 1. Also, since A/B isfinite-by divisible, it is divisible-by-finite and we may assume it is divisible.Let γ ∈ Γ and α be a p -adic integer such that x γ = x α for all x ∈ B .Consider the endomorphism γ − α of A and note that B ≤ ker( γ − α ). Thus A/ ker( γ − α ) ≃ im( γ − α ) is both divisible and reduced, hence trivial. Itfollows γ = α on the whole A . (cid:3) Proof of Theorem 2.2 For the sufficiency of the condition note thatfor any subgroup H ≤ A we have H ∼ H ∩ A and the latter is in turncommensurable with a Γ-subgroup since Γ has bp on A by Theorem 2.1.Concerning necessity, we first prove the statement when A is periodic.Let A = D × R , where D is divisible and R is reduced. Then there is asubgroup R = R Γ ∼ R . Thus DR and D ∩ R are Γ-subgroups of A withfinite index and order respectively. Then we can assume A = D × R . Let X := { a ∈ A | h a i Γ is finite } . Clearly D ≤ X , as Γ has p on D by Lemma8.5. On the other hand, X ∩ R has finite index in R by Lemma 2.8. It follows A/X is finite and by Lemma 2.6 and Theorem 2.1 the statement holds.In the non-periodic case, note that if V is a free subgroup of A such that A/V is periodic, then there is V = V Γ1 ∼ V . Let n := | V / ( V ∩ V ) | . Thusby applying Lemma 2.5 to the Γ-subgroup V := V n we have- there is a free abelian Γ -subgroup V such that A/V is periodic and each γ ∈ Γ acts on V by either the identity or the inversion map .Suppose that V has finite rank. Consider now the action of Γ on theperiodic group A/V and apply the above. Then there is a series V ≤ A ≤ A ≤ A such that A /V and A/A are finite and Γ has either p or ˜ p on A /A .Since A has finite torsion subgroup T we can factor out T and assume A = V . Then Γ has either p or ˜ p on A as straightforward verificationshows.Suppose finally that V has infinite rank. Let V ≤ V be such that V /V is divisible periodic and its p -component has infinite rank for each prime p .We may assume V := V . By the above case when A is periodic, there is aΓ-series V ≤ A ≤ A ≤ A such that A /V and A/A are finite and Γ has p on A /A . We may factor out the torsion subgroup of A , as it is finite, andassume A = V .Again let V ≤ V be such that V /V is divisible periodic and its p -component has infinite rank for each prime p . Let γ ∈ Γ and, for each prime p , let α p be a p -adic integer such that x γ = x α p for all x in the p -componentof A /V . Let ǫ = ± x γ = x ǫ for all x ∈ V . By Lemma 2.5, γ has p on the maximum divisible subgroup D p /V of the p -component of A /V . Thus α p = ǫ on D p /V . Therefore x γ = x ǫ for all x ∈ V and forall x ∈ A /V . We claim that a γ = a ǫ for each a ∈ A . To see this, forany a ∈ A consider n ∈ N such that a n ∈ V . Then there is v ∈ V suchthat a γ = a ǫ v . Hence a nǫ = ( a n ) γ = ( a γ ) n = ( a ǫ v ) n = a nǫ v n . Thus v n = 1.Therefore, as V is torsion-free, we have v = 1, as whished. (cid:3) Recall that locally finite cf -groups are abelian-by-finite and bcf (see [1]). Proof of Theorem B It follows from Proposition 1.1 and Proposition 3.1below. (cid:3) roposition 3.1 Let G be an abelian-by-finite group.i) if G is cn , then G is cf ;ii) if G is bcn , then G is bcf . Proof. Let A be a normal abelian subgroup with finite index r . Then each H ≤ A has at most r conjugates in G . If δ G ( H ) ≤ n < ∞ then for each g ∈ G we have | H : ( H ∩ H g ) | ≤ δ G ( H ) ≤ n hence | H/H G | ≤ (2 n ) r . Moregenerally, if H is any subgroup of G , then | H/H G | ≤ r (2 n ) r . (cid:3) Let us characterize bcf -groups among abelian-by-finite cf -groups. Proposition 3.2 Let G be a non-periodic group with an abelian normal sub-group A with finite index. Then the following are equivalent:i) G is a bcf -group;ii) G is a cf -group and there is B ≤ A such that B has finite exponent, B ⊳ G and each g ∈ G acts by conjugation on A/B by either the identity orthe inversion map. Proof. Let T be the torsion subgroup of A . By Lemma 2.5, for each g ∈ G there ǫ g = ± γ acts on A/T as the automorphism x x ǫ g . Thenthe equivalence of (i) and (ii) holds with B := h A g − ǫ g | g ∈ G i , by Theorem3 of [4]. (cid:3) To prove Theorem A, our first step is a reduction to nilpotent groups. Lemma 3.3 A soluble p -group G with the property cn is nilpotent-by-finite. Proof. By Theorem 2.2, one may refine the derived series of G to a finite G -series S such that G has p on each infinite factor of S . Recall that a p -group of power automorphisms of an abelian p -group is finite (see [11]).Then the stability group S ≤ G of the series S , that is, the intersection ofthe centralizers in G of the factors of the series, has finite index in G . Onthe other hand, by a theorem of Ph.Hall, S is nilpotent. (cid:3) We recall now an elementary property of nilpotent groups. Lemma 3.4 Let G be a nilpotent group with class c . If G ′ has finite exponent e , then G/Z ( G ) has finite exponent dividing e c . roof. Argue by induction on c , the statement being clear for c = 1.Assume c > G/Z has exponent dividing e c − , where Z/γ c ( G ) := Z ( G/γ c ( G )). Then for all g, x ∈ G we have [ g e c − , x ] ∈ γ c ( G ) ≤ G ′ ∩ Z ( G ) . Therefore 1 = [ g e c − , x ] e = [ g e c , x ] , and g e c ∈ Z ( G ), as claimed. (cid:3) Next lemma follows easily from Lemma 6 in [9]. Lemma 3.5 Let G be a nilpotent p-group and N a normal subgroup suchthat G/N is an infinite elementary abelian group. If H and U are finitesubgroup of G such that H ∩ U = 1 , there exists a subgroup V of G such that U ≤ V , H ∩ V = 1 and V N/N is infinite. (cid:3) We deduce a technical lemma which is a tool for our pourpose. Lemma 3.6 Let G be a nilpotent p-group and N be a normal subgroup suchthat G/N is an infinite elementary abelian group. If N contains the fc -center of G and G ′ is abelian with finite exponent, then there are subgroups H , U of G such that H ∩ U = 1 , with injective maps n h n ∈ H and ( i, n ) u i,n ∈ [ G, h − i h n ] ∩ U , where i, n ∈ N , i < n . Proof. Let us show that for each n ∈ N there is an ( n + 1)-uple v n :=( h n , u ,n , u ,n , . . . , u n − ,n ) of elements of G such that:1. { h , . . . , h n } is linearly independent modulo N ;2. u i,n ∈ [ G, h − i h n ] ∀ i ∈ { , . . . , n − } ;3. { u j,h | ≤ j < k ≤ n } is Z -independent in G ′ ;4. H n ∩ U n = 1, where H n := h h , . . . , h n i and U n := h u j,h | ≤ j < k ≤ n i .Then the statement is true for H := S n ∈ N H n and U := S n ∈ N U n .Let h := 1 and choose h ∈ G \ N . Since N contains the fc -center F of G , we have that h has an infinite numbers of coniugates in G , hence [ G, h ]is infinite and residually finite. Thus we may choose u , ∈ [ G, h ] such that h u , i ∩ h h i = 1.Assume then that we have defined v i for i ≤ n , that is, we have elements h , . . . , h n , u j,k , with 0 ≤ j < k ≤ n such that conditions 1-4 hold. To definean adequate v n +1 , note that by Lemma 3.5 we have that there exists V n ≤ G such that H n ≤ V n , U n ∩ V n = 1 and V n N/N is infinite. Then choose11 ) h n +1 ∈ V n \ N U n H n . Note that h n +1 F H n ≤ N H n , so that { h , . . . , h n +1 } is independent mod F .In particular ∀ i ∈ { , . . . , n } , h − i h n +1 F , hence also [ G, h − i h n +1 ] is infinite.Since G ′ is residually finite, we may recursively choose u ,n +1 , . . . , u n,n +1 suchthat ∀ i ∈ { , . . . , n } ii ) u i,n ∈ [ G, h − i h n ] iii ) h u i,n +1 i ∩ U n h u h,n +1 | ≤ h < i i H n +1 = 1Then properties 1-3 hold for v n +1 . Finally suppose there are h ∈ H n , u ∈ U n , s, t , . . . , t n ∈ Z such that iv ) a = hh sn +1 = uu t ,n +1 · · · u t n n,n +1 ∈ H n +1 ∩ U n +1 . Then from ( iii ) it follows u t n n,n +1 = . . . = u t ,n +1 = 1. Hence a = hh sn +1 = u ∈ V n ∩ U n = 1 and 4 holds. (cid:3) Lemma 3.7 Let G be a nilpotent p -group. If G is cn , then G ′ has finiteexponent. Proof. If, by contradiction, G ′ has infinite exponent, then the same happensto the abelian group G ′ /γ ( G ) and there is N such that G ′ ≥ N ≥ γ ( G )and G ′ /N is a Pr¨ufer group. We may assume N = 1, that is, G ′ itself isa Pr¨ufer group and G ′ ≤ Z ( G ). Let us show that for any H ≤ G we have | H G : H | < ∞ , hence G ′ is finite, a contradiction. In fact we have that, bythe cn -property, there is K ⊳ G such that K ∼ H . Thus H has finite indexin HK and we can also assume H = HK , that is, H/H G is finite. Thus, wecan assume H G = 1 and H ∩ G ′ = 1, that is, H is finite with order p n and HG ′ is an abelian Chernikov group. It follows that H is contained in the n -th socle S of HG ′ ⊳ G , where S is finite and normal in G , as whished. (cid:3) Lemma 3.8 Let G be a nilpotent p -group. If G is cn , then G is finite-by-abelian-by-finite. Proof. Let G be a counterexample. Then both G ′ and G/Z ( G ) are infinite.However, they have finite exponent by Lemmas 3.7 and 3.4. Moreover, byLemma 2.6, the fc -center F of G is finite-by-abelian. Thus F has infiniteindex in G . On the other hand, G/F has finite exponent, since F ≥ Z ( G ).Then N := F G p G ′ has infinite index in G , otherwise the abelian group G/F G ′ has finite rank and finite exponent, hence it is finite. This impliesthat the nilpotent group G/F is finite, a contradiction.12f G ′ is abelian we are in a condition to apply Lemma 3.6 and get infiniteelements and subgroups h n ∈ H , u i,n ∈ U as in that statement. By cn -property there is K such that H ∼ K ⊳ G . So that the set { h n ( H ∩ K ) / n ∈ N } is finite. Hence there is i ∈ N and an infinite set I ⊆ N \ { , . . . , i } suchthat for each n ∈ I we have h − i h n ∈ H ∩ K and u i,n ∈ U ∩ [ G, H ∩ K ] ≤ U ∩ K .Therefore U ∩ K is infinite, in contradiction with U ∩ K ∼ U ∩ H = 1.For the general case, proceed by induction on the nilpotency class c > G and assume that the statement is true for G/Z ( G ) and even that thisis finite-by-abelian. Then there is a subgroup L ≤ G such that G/L isabelian and L/Z ( G ) is finite. Thus L ′ is finite and, by the above, G/L ′ isfinite-by-abelian-by-finite, a contradiction. (cid:3) Let us consider now non-periodic cn -groups. Lemma 3.9 Let G be a cn -group and A = A ( G ) its subgroup generated byall infinite cyclic normal subgroups. Then G/A is periodic, A is abelian andeach g ∈ G acts on A by either the identity or the inversion map, hence | G/C G ( A ) | ≤ . Proof. For any x ∈ G there is N ⊳ G which is commensurable with h x i .Then n := | N : ( N ∩ h x i ) | is finite. Thus N n ! ≤ h x i where N n ! ⊳ G . Hence G/A is periodic.It is clear that A is abelian. Let g ∈ G . If h a i ⊳ G and a has infinite order,then there is ǫ a = ± a g = a ǫ a . On the other hand, by Lemma 2.5,there is ǫ = ± a ∈ A there is a periodic element t a ∈ A such that a g = a ǫ t a . It follows a ǫ a − ǫ = t . Therefore ǫ a = ǫ is independent of a , as wished. (cid:3) Proof of Theorem A . Recall from the Introduction that all subgroups of G are subnormal-by-finite. If G is periodic, then, by above quoted resultsin [7] and [3] respectively, we may assume that G is locally nilpotent andsoluble. Then, by Proposition 2.4, only finitely many primary componentsare non-abelian. Thus we may assume G is a p -group and apply Lemma 3.3and Lemma 3.8. It follows that G is finite-by-abelian-by-finite.To treat the general case, consider A = A ( G ) as in Lemma 3.9. We mayassume A is central in G . Let V be a torsion-free subgroup of A such that A/V is periodic. Then G/V is locally finite and we may apply the above.Thus there is a series V ≤ G ≤ G ≤ G such that G acts trivially on V , G /G is abelian, while G /V and G/G are finite. Then we can assume13 = G and note that the stabilizer S of the series has finite index. Since S is nilpotent (by Ph.Hall Theorem) we can assume that G = S is nilpotent.If T is the torsion subgroup of G , then V T /T is contained in the center of G/T . Since all factors of the upper central series of G/T are torsion-free wehave G/T is abelian. Thus G ′ ≤ T ∩ G is finite. (cid:3) Proof of Theorem C . If the statement is false, by Theorem A we mayassume there is a counterexample G periodic and not locally finite. Alsowe may assume G is finitely generated and infinite. Let R be the locallyfinite radical of G . By Theorem A again, R is finite-by-abelian-by-finite. ByTheorem B(i), there is a finite subgroup G ⊳ G such that R/G is abelian-by-finite. We may assume G = 1, so that R is abelian-by-finite.We claim that ¯ G := G/R has finite exponent at most ( n + 1)! where n is such that n ≥ δ G ( H ) for each H ≤ G . In fact, for each x ∈ ¯ G , there is¯ N ⊳ ¯ G such that | ¯ N : ( ¯ N ∩ h x i ) | ≤ n . Thus ¯ N n ! ≤ h x i and ¯ N n ! ⊳ G . Hence¯ N n ! = 1 and x n · n ! = 1.By the positive answer (for all exponents) to the Restricted BurnsideProblem, there is a positive integer k such that every finite image of ¯ G hasorder at most k . Since ¯ G is finitely generated, this means that the finiteresidual ¯ K of ¯ G has finite index and is finitely generated as well. Since also¯ G is locally graded (see [8]), we have ¯ K = 1 and ¯ G is finite. Therefore G isabelian-by-finite, a contradiction. (cid:3) References [1] J. T. Buckley, J.C. Lennox, B. H. Neumann, H. Smith, J. Wiegold,Groups with all subgroups normal-by-finite. J. Austral. Math. Soc. Ser.A (1995), no. 3, 384-398.[2] C. 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