Direct products and elementary equivalence of polycyclic-by-finite groups
aa r X i v : . [ m a t h . G R ] A ug Direct products and elementary equivalenceof polycyclic-by-finite groups
C. Lasserre and F. OgerLasserre Cl´ement: Institut Fourier - U.M.R. 5582, 100 rue des Maths, BP74, 38402 Saint Martin d’H`eres, France; [email protected] Francis: U.F.R. de Math´ematiques, Universit´e Paris 7, Bˆatiment SophieGermain, Case 7012, 75205 Paris Cedex 13, France; [email protected].
Abstract.
Generalizing previous results, we give an algebraic characteriza-tion of elementary equivalence for polycyclic-by-finite groups. We use thischaracterization to investigate the relations between their elementary equiva-lence and the elementary equivalence of the factors in their decompositions indirect products of indecomposable groups. In particular we prove that the el-ementary equivalence G ≡ H of two such groups G, H is equivalent to each ofthe following properties: 1) G ×· · ·× G ( k times G ) ≡ H ×· · ·× H ( k times H )for an integer k ≥
1; 2) A × G ≡ B × H for two polycyclic-by-finite groups A, B such that A ≡ B . It is not presently known if 1) implies G ≡ H for anygroups G, H .
1. Characterization of elementary equivalence.
In the present paper, we investigate the elementary equivalence betweenfinitely generated groups and the relations between direct products and ele-mentary equivalence for groups. First we give the relevant definitions.We say that two groups
M, N are elementarily equivalent and we write M ≡ N if they satisfy the same first-order sentences in the language whichconsists of one binary functional symbol.We say that a group M is polycyclic if there exist some subgroups { } = M ⊂ · · · ⊂ M n = M with M i − normal in M i and M i /M i − cyclic for1 ≤ i ≤ n . For any properties P, Q , we say that M is P -by- Q if there existsa normal subgroup N which satisfies P and such that M/N satisfies Q .For each group M , we denote by Z ( M ) the center of M . For each A ⊂ M ,we denote by h A i the subgroup of M generated by A . We write1 ′ = h{ [ x, y ] | x, y ∈ M }i and, for any h, k ∈ N ∗ , M ′ ( h ) = { [ x , y ] · · · [ x h , y h ] | x , y , . . . , x h , y h ∈ M } , × k M = M × · · · × M ( k times M ), M k = (cid:10)(cid:8) x k | x ∈ M (cid:9)(cid:11) and M k ( h ) = { x k · · · x kh | x , . . . , x h ∈ M } .Each of the properties M ′ = M ′ ( h ) and M k = M k ( h ) can be expressed bya first-order sentence. As polycyclic-by-finite groups are noetherian, it followsfrom [16, Corollary 2.6.2] that, for each polycyclic-by-finite group M andeach k ∈ N ∗ , there exists h ∈ N ∗ such that M ′ = M ′ ( h ) and M k = M k ( h ).Actually, the result concerning the property M ′ = M ′ ( h ) was first proved in[15].For each group M and any subsets A, B , we write AB = { xy | x ∈ A and y ∈ B } . If A, B are subgroups of M and if A or B is normal, then AB isalso a subgroup.For each group M and each subgroup S of M such that M ′ ⊂ S , weconsider the isolator I M ( S ) = ∪ k ∈ N ∗ { x ∈ M | x k ∈ S } , which is a subgroupof M . We write Γ( M ) = I M ( Z ( M ) M ′ ) and ∆( M ) = I M ( M ′ ). We alsohave Γ( M ) = I M ( Z ( M )∆( M )). If M is abelian, then ∆( M ) is the torsionsubgroup τ ( M ).For any h, k ∈ N ∗ , there exists a set of first-order sentences which, ineach group M with M ′ = M ′ ( h ), expresses that Γ( M ) k ⊂ Z ( M ) M ′ (resp.∆( M ) k ⊂ M ′ ).During the 2000s, there was a lot of progress in the study of elementaryequivalence between finitely generated groups, with the proof of Tarski’sconjecture stating that all free groups with at least two generators are ele-mentarily equivalent (see [5] and [17]).Anyway, examples of elementarily equivalent nonisomorphic finitely gen-erated groups also exist for groups which are much nearer to the abelianclass. One of them was given as early as 1971 by B.I. Zil’ber in [18].It quickly appeared that the class of polycyclic-by-finite groups wouldbe an appropriate setting for an algebraic characterization of elementaryequivalence. Actually, two elementarily equivalent such groups necessarilyhave the same finite images (see [9, Remark, p. 475]) and, by [4], any class ofsuch groups which have the same finite images is a finite union of isomorphismclasses.Between 1981 and 1991, algebraic characterizations of elementary equiv-alence were given for particular classes of polycyclic-by-finite groups (see [9],211] and [12]). During the 2000s, some new progress was made with theintroduction of quasi-finitely axiomatizable finitely generated groups by A.Nies (see [6, Introduction]). This line of investigation, and in particular theresults of [6], made it possible to obtain the characterization of elementaryequivalence for polycyclic-by-finite groups which is given below.By Feferman-Vaught’s theorem (see [3]), we have G × G ≡ H × H forany groups G , G , H , H such that G ≡ H and G ≡ H . In particular,we have × k G ≡ × k H for each k ∈ N and any groups G, H such that G ≡ H .In [7, p. 9], L. Manevitz proposes the following conjecture: Conjecture.
For any groups
G, H and each integer k ≥
2, if × k G ≡ × k H ,then G ≡ H .L. Manevitz mentions that the conjecture becomes false if we considersets with one unary function instead of groups. He obtains a counterexampleby taking for G the set N equipped with the successor function and for H the disjoint union of two copies of G . We have × k G ∼ = × k H for each integer k ≥
2, but G and H are not elementarily equivalent.It seems likely that the conjecture is also false for groups in general.Anyway, it appears that no counterexample has been given until present.For some classes of groups, we prove that the conjecture is true by showingthat the groups are characterized up to elementary equivalence by invariantswith good behaviour relative to products. We show in this way that the con-jecture is true for abelian groups, using the invariants introduced by Szmielewor Eklof-Fisher (see [2]).Similarly, by [11, Cor., p. 1042], two finitely generated abelian-by-finitegroups G, H are elementarily equivalent if and only if they have the samefinite images. For polycyclic-by-finite groups, and in particular for finitelygenerated abelian-by-finite groups, the last property is true if and only if
G/G n ∼ = H/H n for each n ∈ N ∗ . The conjecture is true for finitely generatedabelian-by-finite groups because, for any such groups G, H and any k, n ∈ N ∗ ,the finite groups G/G n and H/H n are isomorphic if and only if × k ( G/G n ) ∼ =( × k G ) / ( × k G ) n and × k ( H/H n ) ∼ = ( × k H ) / ( × k H ) n are isomorphic.Also, according to [12, Th., p. 173], which answered a conjecture statedin [8], two finitely generated finite-by-nilpotent groups G, H are elementarilyequivalent if and only if G × Z and H × Z are isomorphic. It follows thatManevitz’s conjecture is true for finitely generated finite-by-nilpotent groups(see [12, Cor. 1, p. 180]). 3n the more general case of polycyclic-by-finite groups, neither of theproperties above is a characterisation of elementary equivalence. Actually(see [12, p. 173]), there exist:1) two finitely generated torsion-free nilpotent groups of class 2, G, H , whichhave the same finite images and do not satisfy G × Z ∼ = H × Z (for suchgroups, G × Z ∼ = H × Z implies G ∼ = H );2) two polycyclic abelian-by-finite groups G, H which have the same finiteimages and do not satisfy G × Z ∼ = H × Z .It follows from [11, Cor., p. 1042] that G and H are elementarily equivalentin the second case and from [12, Th., p. 173] that they are not elementarilyequivalent in the first case.However, Theorem 1.1 below gives an algebraic characterization of ele-mentary equivalence for polycyclic-by-finite groups. In the second section ofthe present paper, we use it in order to prove that the conjecture is true forsuch groups. Theorem 1.1.
Two polycyclic-by-finite groups
G, H are elementarily equiv-alent if and only if there exist n ∈ N ∗ such that | G/G n | = | H/H n | and, foreach k ∈ N ∗ , an injective homomorphism f k : G → H such that H = f k ( G ) Z ( H n ) and | H : f k ( G ) | is prime to k .The following Proposition implies that the case n = 1 of the characteri-zation above is equivalent to the stronger property G × Z ∼ = H × Z . We shallgive the proof of the Proposition first, since it is much shorter than the proofof the Theorem. Proposition 1.2.
For any groups
G, H with
G/G ′ and H/H ′ finitely gen-erated, the following properties are equivalent:1) G × Z ∼ = H × Z ;2) for each k ∈ N ∗ , there exists an injective homomorphism f k : G → H with H = f k ( G ) Z ( H ) and | H/f k ( G ) | prime to k ;3) the same property is true for k = | Γ( H ) / ( Z ( H )∆( H )) | . | ∆( H ) /H ′ | . Proof of Proposition 1.2.
First we show that 1) implies 2). We consideran isomorphism f : M → N with M = G × h u i , N = H × h w i and h u i , h w i infinite. The restriction of f to G ∩ f − ( H ) is an isomorphism from G ∩ f − ( H ) to f ( G ) ∩ H .Suppose f ( G ) = H . Then G/ ( G ∩ f − ( H )) ∼ = h G, f − ( H ) i /f − ( H ) ⊂ M/f − ( H ) ∼ = N/H is infinite cyclic. The same property is true for H/ ( f ( G ) ∩ ). We consider v ∈ G such that G = h v, G ∩ f − ( H ) i and x ∈ H such that H = h x, f ( G ) ∩ H i .We write f ( u ) = w k x l y and f ( v ) = w m x n z with k, l, m, n ∈ Z and y, z ∈ f ( G ) ∩ H . We have u ∈ Z ( M ), whence f ( u ) ∈ Z ( N ) and x l y ∈ Z ( H ). Theintegers l, n are prime to each other since f induces an isomorphism from M/ ( G ∩ f − ( H )) to N/ ( f ( G ) ∩ H ).For each r ∈ Z , we consider the homomorphism f r : G → H defined by f r ( v ) = ( x l y ) r x n z and f r ( t ) = f ( t ) for t ∈ G ∩ f − ( H ). We have f r ( t ) ∈ f ( t ) Z ( H ) for each t ∈ G . It follows N = f ( G ) Z ( N ) = f r ( G ) Z ( N ) and H = f r ( G ) Z ( H ). We also have f r ( v ) ∈ x lr + n ( f ( G ) ∩ H ) since f ( G ) ∩ H isnormal in H . Consequently, f r is injective.For each s ∈ N ∗ , we can choose r in such a way that lr + n is prime to s ,since l and n are prime to each other. Then | H/f r ( G ) | is prime to s .Now it suffices to prove that 3) implies 1). The property H = f k ( G ) Z ( H )implies H ′ = f k ( G ) ′ = f k ( G ′ ). Consequently, | ∆( H ) /f k (∆( G )) | both divides | ∆( H ) /H ′ | since H ′ ⊂ f k (∆( G )) and | H/f k ( G ) | since f k (∆( G )) = ∆( H ) ∩ f k ( G ). As | H/f k ( G ) | is prime to | ∆( H ) /H ′ | , it follows f k (∆( G )) = ∆( H ).Now the property G × Z ∼ = H × Z follows from [12, Prop. 3] and itsproof. Actually, the group G considered in that proposition is supposedfinitely generated finite-by-nilpotent, but the proof only uses the property“ G/ ∆( G ) finitely generated”. (cid:4) The definitions and the two lemmas below will be used in the proof ofTheorem 1.1.For each group M , we write E ( M ) = ∩ k ∈ N ∗ M k . If M is abelian and if τ ( M ) is finite, then E ( M ) is the additive structure of a Q -vector space. Definitions.
Let M be a group, let K be a group of automorphisms of M andlet A, B be subgroups of M stabilized by K such that A ∩ B = [ A, B ] = { } .We say that B is a K -complement of A in M if AB = M , and a K -quasi-complement of A in M if | M : AB | is finite. Lemma 1.3.
Let M be a finitely generated abelian group with additivenotation, let N, S be subgroups of M such that N ∩ S = { } , and let K bea finite group of automorphisms of M which stabilizes N and S . Then thereexists a K -quasi-complement T of N in M which contains S . Proof of Lemma 1.3.
We consider an integer r ≥ r.τ ( M ) = { } and r.I M ( S ) ⊂ S . We write M ∗ = rM , N ∗ = M ∗ ∩ N and S ∗ = M ∗ ∩ S . Then5 ∗ and M ∗ /S ∗ are torsion-free. We have N ∗ ∩ S ∗ = { } and M ∗ , N ∗ , S ∗ are stabilized by K . As M/M ∗ is finite, it suffices to show that N ∗ has a K -quasi-complement in M ∗ which contains S ∗ , or equivalently that N ∗ hasa K -quasi-complement in M ∗ /S ∗ .Consequently, it suffices to prove the Lemma for M finitely generatedtorsion-free abelian and S = { } . Then M can be embedded in a Q -vectorspace M of finite dimension such that the action of K on M is defined.Moreover, K stabilizes the isolator N of N in M , which is a subspace of M .It follows from the proof of Maschke’s theorem given in [1, pp. 226-227] that N has a K -complement T in M . Then T ∩ M is a K -quasi-complement of N in M . (cid:4) Lemma 1.4.
Let M be a torsion-free abelian group with additive notation,let N be a divisible subgroup of M , let S be a subgroup of M such that N ∩ S = { } , and let K be a finite group of automorphisms of M whichstabilizes N and S . Then there exists a K -complement T of N in M whichcontains S . Proof of Lemma 1.4.
It suffices to show that M = N ⊕ S if S is maximal forthe properties N ∩ S = { } and S stabilized by K . Then we have I M ( S ) = S since N ∩ I M ( S ) = { } and I M ( S ) is stabilized by K . We reduce the proofto the case S = { } by considering M/S instead of M .Now suppose S = { } and consider x ∈ M − N . Then K stabilizes T = h{ f ( x ) | f ∈ K }i . In order to obtain a contradiction, it suffices toprove that there exists a nontrivial subgroup U of T stabilized by K suchthat N ∩ U = { } . But, as T is finitely generated, it follows from Lemma1.3 that N ∩ T has a K -quasi-complement U in T . Moreover, U is nontrivialsince T / ( N ∩ T ) is torsion-free and therefore infinite. (cid:4) Proof of Theorem 1.1.
First we show that the condition is necessary.We consider a finite generating tuple x of G . By [6, Th. 3.1], there existan integer n ≥ k ≥
1, a formula ϕ k ( u ) satisfiedby x in G such that, for each finitely generated group M and each tuple y which satisfies ϕ k in M , the map x → y induces an isomorphism from G/Z ( G n ) to M/Z ( M n ) and an injective homomorphism h : G → M with | M : h ( G ) | prime to k . As G and H are elementarily equivalent, we have | G/G n | = | H/H n | and there exists a tuple y which satisfies ϕ k in H .It remains to be proved that the condition is sufficient. First we show6ome properties which are true for each injective homomorphism f : G → H such that H = f ( G ) Z ( H n ).The homomorphism f induces an isomorphism from G/G n to H/H n since H = f ( G ) H n and | G/G n | = | H/H n | . Consequently, we have f − ( H n ) = G n , f ( G n ) = H n ∩ f ( G ) and H n = ( H n ∩ f ( G )) Z ( H n ) = f ( G n ) Z ( H n ). It follows( H n ) ′ = ( f ( G n )) ′ = f (( G n ) ′ ) and f − (∆( H n )) = ∆( G n ).Now we show that f − ( Z ( H n )) = Z ( G n ). For each x ∈ G such that f ( x ) ∈ Z ( H n ), we have x ∈ G n since f ( x ) ∈ H n , and there exists no y ∈ G n such that [ x, y ] = 1 since it would imply f ( y ) ∈ H n and [ f ( x ) , f ( y )] = 1.Conversely, for each x ∈ Z ( G n ), we have f ( x ) ∈ H n . Let us suppose thatthere exists z ∈ H n such that [ f ( x ) , z ] = 1. Then we have z = z n · · · z nr with r ∈ N ∗ and z , . . . , z r ∈ H . For each i ∈ { , . . . , r } , we consider y i ∈ G such that z i ∈ f ( y i ) Z ( H n ). As Z ( H n ) is normal in H , we have z ∈ f ( y ) n · · · f ( y r ) n Z ( H n ), whence f ([ x, y n · · · y nr ]) = [ f ( x ) , f ( y ) n · · · f ( y r ) n ] =[ f ( x ) , z ] = 1 and [ x, y n · · · y nr ] = 1, which contradicts x ∈ Z ( G n ).Consequently, we have f ( Z ( G n )) = Z ( H n ) ∩ f ( G ) and f induces an iso-morphism from G/Z ( G n ) to H/Z ( H n ). It follows | H : f ( G ) | = | Z ( H n ) /f ( Z ( G n )) | .Now suppose that | H : f ( G ) | is prime to | ∆( H n ) / ( H n ) ′ | . Then f − (∆( H n )) =∆( G n ) implies f (∆( G n )) = ∆( H n ) since | ∆( H n ) /f (∆( G n )) | both divides | ∆( H n ) / ( H n ) ′ | and | H : f ( G ) | . Moreover, the equalities f ( Z ( G n )) = Z ( H n ) ∩ f ( G ) and f (∆( G n )) = ∆( H n ) imply f ( Z ( G n ) ∩ ∆( G n )) = Z ( H n ) ∩ ∆( H n ).For each k ∈ N , we consider an injective homomorphism f k : G → H with H = f k ( G ) Z ( H n ) and | H : f k ( G ) | prime to l ! where l = sup( k, | ∆( H n ) / ( H n ) ′ | ).Each f k satisfies f − k ( H n ) = G n , f − k ( Z ( H n )) = Z ( G n ), H n = f k ( G n ) Z ( H n ), f k (∆( G n )) = ∆( H n ), f k ( Z ( G n ) ∩ ∆( G n )) = Z ( H n ) ∩ ∆( H n ) and | H : f k ( G ) | = | Z ( H n ) /f k ( Z ( G n )) | .We consider an ω -incomplete ultrafilter U over N and the injective ho-momorphism f : G U → H U which admits ( f k ) k ∈ N as a representative.We have ( G U ) n = ( G n ) U and ( H U ) n = ( H n ) U since there exists an integer r ≥ G n = G n ( r ) and H n = H n ( r ). Consequently, f inducesan injective homomorphism from ( G U ) n to ( H U ) n and an isomorphism from G U / ( G U ) n to H U / ( H U ) n .We also have (( G U ) n ) ′ = (( G n ) U ) ′ = (( G n ) ′ ) U and (( H U ) n ) ′ = (( H n ) U ) ′ =(( H n ) ′ ) U since there exists an integer s ≥ G n ) ′ = ( G n ) ′ ( s )and ( H n ) ′ = ( H n ) ′ ( s ). It follows ∆(( G U ) n ) = ∆( G n ) U and ∆(( H U ) n ) =∆( H n ) U since there exists an integer t ≥ G n ) t ⊂ ( G n ) ′ and∆( H n ) t ⊂ ( H n ) ′ . Now the equalities f k (∆( G n )) = ∆( H n ) for k ∈ N imply f (∆(( G U ) n )) = ∆(( H U ) n ). 7e have Z (( G U ) n ) = Z (( G n ) U ) = Z ( G n ) U and Z (( H U ) n ) = Z ( H n ) U .The equalities f k ( Z ( G n ) ∩ ∆( G n )) = Z ( H n ) ∩ ∆( H n ) imply f ( Z (( G U ) n ) ∩ ∆(( G U ) n )) = Z (( H U ) n ) ∩ ∆(( H U ) n ).The action of G/G n on Z ( G n ) is defined since G n acts trivially on Z ( G n ),and the action of G U / ( G U ) n on Z (( G U ) n ) is defined since ( G U ) n acts triviallyon Z (( G U ) n ). The same properties are true for H .According to Lemma 1.3, there exist a G/G n -quasi-complement C of Z ( G n ) ∩ ∆( G n ) in Z ( G n ) and an H/H n -quasi-complement D of Z ( H n ) ∩ ∆( H n ) in Z ( H n ).Then C U is a G U / ( G U ) n -quasi-complement of Z (( G U ) n ) ∩ ∆(( G U ) n ) in Z (( G U ) n ), and E ( C U ) is a G U / ( G U ) n -complement of E ( Z (( G U ) n ) ∩ ∆(( G U ) n ))in E ( Z (( G U ) n )) since E ( Z (( G U ) n )) is a Q -vector space. In particular, E ( C U )is a normal subgroup of G U . The same properties are true for H and D .Now we prove that there exists a subgroup T of G U such that G ⊂ T , T ∩ E ( C U ) = { } and G U = T.E ( C U ). As E ( C U ) ∩ ∆(( G U ) n ) = { } , itsuffices to show that there exists a subgroup S of G U / ∆(( G U ) n ) such that G/ ∆( G ) ⊂ S , S ∩ E ( C U ) = { } and S.E ( C U ) = G U / ∆(( G U ) n ).The action of G U / ( G U ) n on ( G U ) n / ∆(( G U ) n ) is defined since ( G U ) n / ∆(( G U ) n )is an abelian normal subgroup of G U / ∆(( G U ) n ). By Lemma 1.4, E ( C U ) hasa G U / ( G U ) n -complement R in ( G U ) n / ∆(( G U ) n ) which contains G n / ∆( G n ).Let us consider x , . . . , x m ∈ G such that G is the disjoint union of x G n , . . . , x m G n . Then S = x R ∪ . . . ∪ x m R is a subgroup of G U / ∆(( G U ) n )which contains G/ ∆( G n ). We have S ∩ E ( C U ) = { } and S.E ( C U ) = G U / ∆(( G U ) n ).For each k ∈ N , f k induces an isomorphism from Z ( G n ) /Z ( G n ) k ! to Z ( H n ) /Z ( H n ) k ! since | Z ( H n ) /f k ( Z ( G n )) | = | H : f k ( G ) | is prime to k !. Con-sequently, f induces an isomorphism from Z (( G U ) n ) /E ( Z (( G U ) n )) to Z (( H U ) n ) /E ( Z (( H U ) n )), and therefore induces an isomorphism from G U /E ( Z (( G U ) n )) to H U /E ( Z (( H U ) n )) since it induces an isomorphism from G U /Z (( G U ) n ) to H U /Z (( H U ) n ).It follows that f induces an isomorphism from G U /E ( C U ) to H U /E ( D U )since f ( E ( Z (( G U ) n ) ∩ ∆(( G U ) n ))) = E ( Z (( H U ) n ) ∩ ∆(( H U ) n )). In particular,each element of H U can be written in a unique way as a product of an elementof f ( T ) and an element of E ( D U ).Now denote by u , . . . , u m the elements of G U / ( G U ) n and write v i = f ( u i )for 1 ≤ i ≤ m . The restriction of f to T is completed into an isomorphismfrom G U to H U by any isomorphism g : E ( C U ) → E ( D U ) which satisfies g ( z u i ) = g ( z ) v i for z ∈ E ( C U ) and 1 ≤ i ≤ m . It remains to be proved that8 exists.The automorphisms θ u i : x → x u i of Z ( G n ) / ( Z ( G n ) ∩ ∆( G n )) and theautomorphisms θ v i : x → x v i of Z ( H n ) / ( Z ( H n ) ∩ ∆( H n )) are defined since G n acts trivially on Z ( G n ) and H n acts trivially on Z ( H n ).Each f k induces an injective homomorphism from A = ( Z ( G n ) / ( Z ( G n ) ∩ ∆( G n )) , θ u , . . . , θ u m ) to B = ( Z ( H n ) / ( Z ( H n ) ∩ ∆( H n )) , θ v , . . . , θ v m )and therefore induces an isomorphism from A/A k ! to B/B k ! since | Z ( H n ) /f k ( Z ( G n )) | = | H : f k ( G ) | is prime to k !.By [13, Cor. 1.2], A and B are elementarily equivalent. Moreover, theexistence of an isomorphism from A U to B U follows from the proofs of [11]and [13], and from the existence of isomorphisms b Z U → Z U → b Z × E ( Z U )which fix the elements of Z , where b Z is the profinite completion of Z .Any isomorphism from A U = ( Z (( G U ) n ) / ( Z (( G U ) n ) ∩ ∆(( G U ) n )) , θ u , . . . , θ u m ) to B U = ( Z (( H U ) n ) / ( Z (( H U ) n ) ∩ ∆(( H U ) n )) , θ v , . . . , θ v m )induces an isomorphism from E ( A U ) to E ( B U ), and therefore induces anisomorphism g : E ( C U ) → E ( D U ) which satisfies g ( z u i ) = g ( z ) v i for z ∈ E ( C U ) and 1 ≤ i ≤ m . (cid:4)
2. Direct products and elementary equivalence.
For each group G and any subgroups A, B , we write G = A × B if G = AB and A ∩ B = [ A, B ] = { } . Theorem 2.1.
Consider a polycyclic-by-finite group G , an integer k ≥ G ) k ⊂ Z ( G )∆( G ) and ∆( G ) k ⊂ G ′ , an integer n ≥ x , . . . , x n ⊂ G such that G = h x i × · · · × h x n i . Write G i = h x i i for 1 ≤ i ≤ n . Then there exist an integer r ≥ m ≥
1, a formula ϕ m ( u , . . . , u n ) satisfied by ( x , . . . , x n ) in G suchthat, for each finitely generated group H with Γ( H ) k ⊂ Z ( H )∆( H ) and∆( H ) k ⊂ H ′ , and for each ( y , . . . , y n ) which satisfies ϕ m in H , there existsome subgroups H i ⊂ C H ( y , . . . , y i − , y i +1 , . . . , y n ) with | H i /H ri | = | G i /G ri | such that H = H × · · · × H n and such that the maps x i → pr H i ( y i ) in-duce injective homomorphisms f i : G i → H i with H i = f i ( G i ) Z ( H ri ) and | H i : f i ( G i ) | prime to m . Proof.
By [6, Th. 3.1], there exist an integer r ≥ m ≥
1, a formula ψ m ( u , . . . , u n ) satisfied by ( x , . . . , x n ) in G such that, for9ach finitely generated group H and each ( u , . . . , u n ) which satisfies ψ m in H , the map ( x , . . . , x n ) → ( u , . . . , u n ) induces an injective homomorphism f : G → H with H = f ( G ) Z ( H r ) and | H : f ( G ) | prime to m .It suffices to show that ϕ m exists for m divisible by r . We fix m for theremainder of the proof.For 1 ≤ i ≤ n , we write x ∗ i = ( x , . . . , x i − , x i +1 , . . . , x n ) and u ∗ i =( u , . . . , u i − , u i +1 , . . . , u n ). We have G i ⊂ C G ( x ∗ i ) = Z ( G ) ×· · ·× Z ( G i − ) × G i × Z ( G i +1 ) ×· · ·× Z ( G n ) = G i Z ( G ).It follows that G = C G ( x ∗ ) · · · C G ( x ∗ n ).We consider an integer t ≥ G ′ = G ′ ( t ). We denote by α ( u , . . . , u n ) a formula which says that H ′ = H ′ ( t ), [ C H ( u ∗ i ) , C H ( u ∗ j )] = { } for i = j and H = C H ( u ∗ ) · · · C H ( u ∗ n ). It follows that Z ( C H ( u ∗ i )) = Z ( H ) for1 ≤ i ≤ n and H/Z ( H ) = C H ( u ∗ ) /Z ( H ) × · · · × C H ( u ∗ n ) /Z ( H ). The formula α is satisfied by ( x , . . . , x n ) in G .From now on, we only consider groups H such that H ′ = H ′ ( t ), Γ( H ) k ⊂ Z ( H )∆( H ) and ∆( H ) k ⊂ H ′ , and sequences ( u , . . . , u n ) which satisfy α ∧ ψ m in H . For each i ∈ { , . . . , n } , we consider a sequence of terms ρ i ( u i ) suchthat each element of ∆( G i ) can be written in a unique way in the form xy with x ∈ ρ i ( x i ) and y ∈ G ′ i .As α ( u , . . . , u n ) implies C H ( u ∗ i ) ′ = C H ( u ∗ i ) ′ ( t ) for 1 ≤ i ≤ n , there existsa formula β ( u , . . . , u n ) satisfied by ( x , . . . , x n ) in G which expresses that:1) H ′ = C H ( u ∗ ) ′ × · · · × C H ( u ∗ n ) ′ (it suffices to say that { } = C H ( u ∗ i ) ′ ∩ ( C H ( u ∗ ) ′ · · · C H ( u ∗ i − ) ′ C H ( u ∗ i +1 ) ′ · · · C H ( u ∗ n ) ′ ) for 1 ≤ i ≤ n );2) vw ∈ ρ i ( u i ) C H ( u ∗ i ) ′ for 1 ≤ i ≤ n and v, w ∈ ρ i ( u i );3) each element of ∆( H ) can be written in a unique way as v · · · v n v with v ∈ H ′ and v i ∈ ρ i ( u i ) for 1 ≤ i ≤ n (here we use ∆( H ) k ⊂ H ′ ).It follows from 1), 2), 3) that ∆( H ) is the direct product of the subgroups ρ i ( u i ) C H ( u ∗ i ) ′ for 1 ≤ i ≤ n .From now on, we only consider sequences ( u , . . . , u n ) which satisfy α ∧ β ∧ ψ m in H .For each i ∈ { , . . . , n } , we consider some terms σ i, ( u i ) , . . . , σ i,r ( i ) ( u i )such that each element of Γ( G i ) can be written in a unique way as σ i, ( x i ) a · · · σ i,r ( i ) ( x i ) a r ( i ) y with a , . . . , a r ( i ) ∈ Z and y ∈ ∆( G i ).There exists a formula γ ( u , . . . , u n ) satisfied by ( x , . . . , x n ) in G whichexpresses that each element of Γ( H ) can be written as u = Π ≤ j ≤ r ( i )1 ≤ i ≤ n σ i,j ( u i ) a i,j v km w with 0 ≤ a i,j ≤ km − u for any i, j , and v ∈ Γ( H ), w ∈ ∆( H ). 10or each i ∈ { , . . . , n } , we also consider some terms τ i, ( u i ) , . . . , τ i,s ( i ) ( u i )such that each element of G i can be written in a unique way as τ i, ( x i ) a · · · τ i,s ( i ) ( x i ) a s ( i ) y with a , . . . , a s ( i ) ∈ Z and y ∈ Γ( G i ). It followsthat each element of C G ( x ∗ i ) = G i Z ( G ) can be written as x = τ i, ( x i ) a · · · τ i,s ( i ) ( x i ) a s ( i ) y m z with 0 ≤ a , . . . , a s ( i ) ≤ m − x , and y ∈ C G ( x ∗ i ), z ∈ Γ( C G ( x ∗ i )).Consequently, there exists a formula δ i ( u , . . . , u n ) satisfied by ( x , . . . , x n )in G which expresses that each element of C H ( u ∗ i ) can be written as u = τ i, ( u i ) a · · · τ i,s ( i ) ( u i ) a s ( i ) v m w with 0 ≤ a , . . . , a s ( i ) ≤ m − u , and v ∈ C H ( u ∗ i ), w ∈ Γ( C H ( u ∗ i )).We denote by δ the conjunction of the formulas δ i . From now on, we onlyconsider sequences ( u , . . . , u n ) which satisfy ϕ m = α ∧ β ∧ γ ∧ δ ∧ ψ m in H .As the abelian group Γ( H ) / ∆( H ) is finitely generated and torsion-free,it is freely generated by the images of a family of elements ( v i,j ) ≤ j ≤ r ( i )1 ≤ i ≤ n ⊂ Γ( H ), where each v i,j can be written as σ i,j ( u i ) v km w with v ∈ Γ( H ) and w ∈ ∆( H ), which implies v km w ∈ Z ( H ) m ∆( H ) since Γ( H ) k ⊂ Z ( H )∆( H ).For any i, j , we choose w in such a way that v km w ∈ Z ( H ) m , which implies v i,j ∈ σ i,j ( u i ) Z ( H ) m ⊂ C H ( u ∗ i ).For each i ∈ { , . . . , n } , as the abelian group C H ( u ∗ i ) / Γ( C H ( u ∗ i )) is finitelygenerated and torsion-free, it is freely generated by the images of a se-quence of elements ( w i,j ) ≤ j ≤ s ( i ) , where each w i,j can be written in the form τ i,j ( u i ) w m with w ∈ C H ( u ∗ i ), which implies w i,j ∈ C H ( u ∗ i ).For each i ∈ { , . . . , n } , we denote by H i the subgroup of H generatedby C H ( u ∗ i ) ′ , ρ i ( u i ), v i, , . . . , v i,r ( i ) , w i, , . . . , w i,s ( i ) . We have H i ⊂ C H ( u ∗ i ) andeach element of H i can be written in a unique way as uv a i, · · · v a r ( i ) i,r ( i ) w b i, · · · w b s ( i ) i,s ( i ) with u ∈ ρ i ( u i ) C H ( u ∗ i ) ′ , a , . . . , a r ( i ) ∈ Z and b , . . . , b s ( i ) ∈ Z .We have H = H × · · · × H n because of the following facts:1) For i = j , [ H i , H j ] = { } follows from [ C H ( u ∗ i ) , C H ( u ∗ j )] = { } .2) We have ∆( H ) = ρ ( u ) C H ( u ∗ ) ′ × · · · × ρ n ( u n ) C H ( u ∗ n ) ′ .3) We have Γ( H ) / ∆( H ) = h v ′ , , . . . , v ′ ,r (1) i × · · · × h v ′ n, , . . . , v ′ n,r ( n ) i wherethe v ′ i,j are the images of the v i,j in H/ ∆( H ).4) The properties Z ( H ) = Z ( C H ( u ∗ )) = · · · = Z ( C H ( u ∗ n )) and H/Z ( H ) = C H ( u ∗ ) /Z ( H ) × · · · × C H ( u ∗ n ) /Z ( H ) imply H/ Γ( H ) = C H ( u ∗ ) / Γ( C H ( u ∗ )) × · · · × C H ( u ∗ n ) / Γ( C H ( u ∗ n ))= h w ′ , , . . . , w ′ ,s (1) i × · · · × h w ′ n, , . . . , w ′ n,s ( n ) i where the w ′ i,j are the images of the w i,j in C H ( u ∗ i ) / Γ( C H ( u ∗ i )).Now we prove that (cid:12)(cid:12) H i : pr H i ( f ( G i )) (cid:12)(cid:12) is prime to m for 1 ≤ i ≤ n .11s ( u , . . . , u n ) satisfies γ ∧ δ in H , f induces some injective homomo-morphisms from G/ Γ( G ) to H/ Γ( H ) and from Γ( G ) / ∆( G ) to Γ( H ) / ∆( H ).It follows f (Γ( G )) = Γ( H ) ∩ f ( G ) and f (∆( G )) = ∆( H ) ∩ f ( G ).Moreover, | ∆( H ) : f (∆( G )) | is prime to m because it divides | H : f ( G ) | = | H : f ( G )∆( H ) | . | ∆( H ) : f (∆( G )) | . It follows that | ∆( H i ) : f (∆( G i )) | isprime to m since ∆( G ) = ∆( G ) ×· · ·× ∆( G n ), ∆( H ) = ∆( H ) ×· · ·× ∆( H n )and f (∆( G j )) ⊂ ∆( H j ) for 1 ≤ j ≤ n .We have v i,j pr H i ( σ i,j ( u i )) − = pr H i ( v i,j σ i,j ( u i ) − ) ∈ pr H i ( H m ) = H mi for1 ≤ j ≤ r ( i ) and w i,j pr H i ( τ i,j ( u i )) − = pr H i ( w i,j τ i,j ( u i ) − ) ∈ pr H i ( H m ) = H mi for 1 ≤ j ≤ s ( i ). Consequently, H i = (cid:10) ∆( H i ) , ( v i,j ) ≤ j ≤ r ( i ) , ( w i,j ) ≤ j ≤ s ( i ) (cid:11) is generated by pr H i ( f ( G i )), H mi and ∆( H i ), and (cid:12)(cid:12) H i : pr H i ( f ( G i ))∆( H i ) (cid:12)(cid:12) isprime to m . It follows that (cid:12)(cid:12) H i : pr H i ( f ( G i )) (cid:12)(cid:12) is prime to m since pr H i ( f (∆( G i ))) = f (∆( G i )) and | ∆( H i ) : f (∆( G i )) | is prime to m .Finally, we show that H i = pr H i ( f ( G i )) Z ( H ri ) for 1 ≤ i ≤ n .We have H ′ ⊂ f ( G ) ′ Z ( H r ) = f ( G ′ ) Z ( H r ) because H = f ( G ) Z ( H r )and Z ( H r ) is normal in H . As f (∆( G )) contains each ρ j ( u j ), it follows∆( H ) ⊂ f (∆( G )) Z ( H r ). Consequently, we have ∆( H i ) ⊂ f (∆( G i )) Z ( H ri ) =pr H i ( f (∆( G i ))) Z ( H ri ) since pr H i (∆( H )) = ∆( H i ), pr H i ( Z ( H r )) = Z ( H ri ),pr H i ( f (∆( G i ))) = f (∆( G i )) and pr H i ( f (∆( G j ))) = { } for j ∈ { , . . . , n } −{ i } .For 1 ≤ j ≤ r ( i ), we have v i,j ∈ σ i,j ( u i ) Z ( H ) m , and therefore v i,j ∈ pr H i ( σ i,j ( u i )) Z ( H i ) m since pr H i ( v i,j ) = v i,j and pr H i ( Z ( H ) m ) = Z ( H i ) m . As r divides m , we have Z ( H i ) m ⊂ Z ( H i ) r ⊂ Z ( H ri ). Consequently,pr H i ( f (Γ( G i ))) Z ( H ri ) contains v i, , . . . , v i,r ( i ) and therefore contains Γ( H i ).For each z ∈ H i , as H = f ( G ) Z ( H r ), there exist x ∈ h x i , . . . , x n ∈h x n i , y ∈ Z ( H r ) such that z = f ( x ) · · · f ( x n ) y . It follows that z =pr H i ( z ) = pr H i ( f ( x )) · · · pr H i ( f ( x n ))pr H i ( y ) belongs to pr H i ( f ( G i )) Z ( H ri )since pr H i ( y ) ∈ Z ( H ri ), pr H i ( f ( x i )) ∈ pr H i ( f ( G i )) and pr H i ( f ( x j )) ∈ Z ( H i ) ⊂ Γ( H i ) ⊂ pr H i ( f (Γ( G i ))) Z ( H ri ) for j ∈ { , . . . , n } − { i } . (cid:4) Theorem 2.2.
Let
G, H be elementarily equivalent polycyclic-by-finitegroups. Then, for each decomposition G ∼ = G × · · · × G m , there existsa decomposition H ∼ = H × · · · × H m with H i ≡ G i for 1 ≤ i ≤ m . Proof.
There exists k ∈ N ∗ such that Γ( G ) k ⊂ Z ( G ) G ′ , ∆( G ) k ⊂ G ′ ,Γ( H ) k ⊂ Z ( H ) H ′ and ∆( H ) k ⊂ H ′ . Consequently, by Theorem 2.1, thereexist r ∈ N ∗ and, for each n ∈ N ∗ , a decomposition H ∼ = H ,n ×· · ·× H m,n and12ome injective homomorphisms g i,n : G i → H i,n with (cid:12)(cid:12) H i,n /H ri,n (cid:12)(cid:12) = | G i /G ri | , H i,n = g i,n ( G i ) Z ( H ri,n ) and | H i,n : g i,n ( G i ) | prime to n ! for 1 ≤ i ≤ m .According to [14, Cor. 3], up to isomorphism, H only has finitely manydecompositions in direct products of indecomposable groups. It follows that,up to isomorphism, H only has finitely many decompositions in direct prod-ucts of groups, since any such decomposition is obtained by grouping togethersome factors of a decomposition in direct product of indecomposable groups.Consequently, there exists S ⊂ N ∗ infinite such that, for each i ∈ { , . . . , m } ,the subgroups H i,n for n ∈ S are all isomorphic. By Theorem 1.1, it follows G i ≡ H i,n for 1 ≤ i ≤ m and n ∈ S . (cid:4) Remark.
In particular, if H is indecomposable, then G is indecomposable.The following definitions are slightly different from those given by F. Ogerin [14]. We are using them because they are easier to manage. Definitions.
Denote by J the infinite cyclic group with multiplicative no-tation. Let G be a group with G/G ′ finitely generated. Then G is J -indecomposable if it is indecomposable and if, for each r ∈ N ∗ and any groups M, N , ( × r J ) × G ∼ = M × N implies M or N torsion-free abelian (in partic-ular, J is J -indecomposable). A J -decomposition of G is a decomposition( × r J ) × G ∼ = G × · · · × G n with G , . . . , G n J -indecomposable.According to [14, Prop. 1], each group M with M/M ′ finitely generatedwhich satisfies the maximal condition on direct factors, and in particulareach polycyclic-by-finite group, has a J -decomposition. By [14, Th.], for any J -indecomposable groups G , . . . , G m , H , . . . , H n with G /G ′ , . . . , G m /G ′ m , H /H ′ , . . . , H n /H ′ n finitely generated, if G × · · · × G m ∼ = H × · · · × H n , then m = n and there exists a permutation σ of { , . . . , n } such that J × G i ∼ = J × H σ ( i ) for 1 ≤ i ≤ n .According to [10], for any groups G, H , if J × G ∼ = J × H , then G ≡ H .By [14, Lemma 1], J × J × G ∼ = J × H implies J × G ∼ = H . Lemma 2.3.
For any elementarily equivalent polycyclic-by-finite groups
G, H , if G is J -indecomposable, then H is J -indecomposable. Proof.
Otherwise, there exists a decomposition ( × r J ) × H ∼ = H × H with H , H not torsion-free abelian. As ( × r J ) × G ≡ ( × r J ) × H , Theorem 2.213mplies the existence of a decomposition ( × r J ) × G ∼ = G × G with G ≡ H and G ≡ H . It follows that G is not J -indecomposable. (cid:4) Corollary 2.4.
For any elementarily equivalent polycyclic-by-finite groups G , H , each r ∈ N ∗ and any J -decompositions ( × r J ) × G ∼ = G × · · · × G m and( × r J ) × H ∼ = H × · · · × H n , we have m = n and there exists a permutation σ of { , . . . , n } such that G i ≡ H σ ( i ) for 1 ≤ i ≤ n . Proof.
As ( × r J ) × G ≡ ( × r J ) × H , Theorem 2.2 implies the existence of adecomposition ( × r J ) × H ∼ = K × · · · × K m with K i ≡ G i for 1 ≤ i ≤ m . ByLemma 2.3, each K i is J -indecomposable. According to [14, Th.], we have m = n and there exists a permutation σ of { , . . . , n } such that J × H σ ( i ) ∼ = J × K i , and therefore H σ ( i ) ≡ K i ≡ G i , for 1 ≤ i ≤ n . (cid:4) Lemma 2.5.
For any polycyclic-by-finite groups
G, H and each r ∈ N ∗ , if( × r J ) × G ≡ ( × r J ) × H , then G ≡ H . Proof.
It follows from Theorem 2.2 applied to ( × r J ) × G and ( × r J ) × H that there exists a group K ≡ G such that ( × r J ) × H ∼ = ( × r J ) × K , whichimplies J × H ∼ = J × K and H ≡ K ≡ G . (cid:4) Corollary 2.6.
For any polycyclic-by-finite groups G , G , H , H such that G ≡ H , we have G × G ≡ H × H if and only if G ≡ H . Proof.
The condition is sufficient by Feferman-Vaught’s theorem. It remainsto be proved that it is necessary.By Corollary 2.4, there exist some J -decompositions ( × r J ) × G ∼ = G , ×· · · × G ,k and ( × r J ) × H ∼ = H , × · · · × H ,k with G ,i ≡ H ,i for 1 ≤ i ≤ k .We also consider some J -decompositions ( × s J ) × G ∼ = G , × · · · × G ,m and( × s J ) × H ∼ = H , × · · · × H ,n .The property G × G ≡ H × H implies ( × r + s J ) × G × G ≡ ( × r + s J ) × H × H . It follows from Corollary 2.4 applied to the J -decompositions( × r + s J ) × G × G ∼ = G , × · · · × G ,k × G , × · · · × G ,m and ( × r + s J ) × H × H ∼ = H , × · · · × H ,k × H , × · · · × H ,n that, for each group K ,we have |{ i ∈ { , . . . , m } | G ,i ≡ K }| = |{ j ∈ { , . . . , n } | H ,j ≡ K }| since |{ i ∈ { , . . . , k } | G ,i ≡ K }| = |{ j ∈ { , . . . , k } | H ,j ≡ K }| . Consequently,we have ( × s J ) × G ≡ ( × s J ) × H . It follows G ≡ H by Lemma 2.5. (cid:4) Corollary 2.7.
For any polycyclic-by-finite groups
G, H and each integer k ≥