aa r X i v : . [ m a t h . G R ] S e p EMBEDDING THEOREMS FOR SOLVABLE GROUPS
VITALY ROMAN’KOV Abstract.
In this paper, we prove a series of results on group embeddingsin groups with a small number of generators. We show that each finitelygenerated group G lying in a variety M can be embedded in a 4-generatedgroup H ∈ MA ( A means the variety of abelian groups). If G is a finitegroup, then H can also be found as a finite group. It follows, that any finitelygenerated (finite) solvable group G of the derived length l can be embeddedin a 4-generated (finite) solvable group H of length l + 1. Thus, we answer thequestion of V. H. Mikaelian and A.Yu. Olshanskii. It is also shown that anycountable group G ∈ M , such that the abelianization G ab is a free abeliangroup, is embeddable in a 2-generated group H ∈ MA . Key words. Solvable group, embedding, variety2020 Mathematical Subject Classification. 20F16, 20E22 Introduction
The main aim of this paper is to study embeddings of finitely generated groupsin 2- or 4-generated groups. Let G stand for a finitely generated group which lies ina variety M , and H for a 4-generated group in which G can be embedded. We showthat H can be found in the variety MA , where A denote the variety of abeliangroups. It follows that every finitely generated solvable group of the derived length l can be embedded in a 4-generated solvable group of length l + 1.We also study what further properties of G our embedding procedure endow H. Finiteness is one of them, and if G is a finite p -group ( p is a prime) , H can bechosen as a finite p -group. Also, if G has finite exponent, then H can be made tohave finite exponent. If G is a counable group such that the abelianization G ab isa free abelian group then H can be found as a 2-generated group.Thus, we refine the classical results on embeddings of countable groups, a briefoverview of which is given below. These results refer to embeddings in 2-generatedgroups. We do not know whether parameter 4 can be lowered in our results. Webelieve that this cannot be done.In the late 1940s, G. Higman, B. H. Neumann and H. Neumann showed thatevery countable group embeds in a 2-generator group, in the same paper [3] in whichthey introduced and succesifully applied HNN-extensions. Their method using freeconstructions of groups did not give similar results for varieties of groups. So thenB. H. Neumann and H. Neumann in [6] applied wreath products to prove that everycountable group G lying in a variety M can be embedded in a 2-generated group H ∈ MA . If G is finite ( p -) group, then H can be chosen as finite ( p -) group.Also, if G has finite exponent, then H can be made to have finite exponent. The work is supported by Mathematical Center in Akademgorodok under agreement No.075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
It follows that every countable solvable group of the derived length l can beembedded in some 2-generated solvable group of length l +2 . Note that this bound issharp. Namely, the group Q of rationals does not embed into any finitely generatedmetabelian group M. Indeed, M is residually finite by Hall ′ s theorem proved in[1], but Q is not. Thus we cannot lower l + 2 to l + 1 in the Neumann-Neumannembedding theorem.V.H. Mikaelian and A. Yu. Olshanskii gave an explicit classification of all abeliangroups that can occur as subgroups of finitely generated metabelian groups asfollows. Theorem (V. H. Mikaelian, A. Yu. Olshanskii [5]). Let A be an abelian group.The following properties are equivalent.(1) A is a subgroup of a finitely generated metabelian group;(2) A is a subgroup of a finitely generated abelian-by-polycyclic group;(3) A is a subgroup of a finitely presented metabelian group;(4) A is a subgroup of a 2-generated metabelian group;(5) A is a Hall group.Note that (3) follows from a remarkable statement independently proved byG. Baumslag and V.N. Remeslennikov: each finitely generated metabelian groupembeds in some finitely presented metabelian group (see [10]).By definition, A is a Hall group if • A is a (finite or) countable abelian group; • A = T ⊕ K, where T is a bounded torsion group (i.e., the orders of allelements in T are bounded), K is torsion free; • K has a free abelian subgroup F such that K/ F is a torsion group withtrivial p -subgroups for all primes except for the members of a finite setdefined by K. In [7], A. Yu. Olshanskii established a number of other embedding theorems formetabelian groups.Every finitely generated nilpotent group satisfies the maximal condition on sub-groups, that is, every subgroup is finitely generated (see [2] or [10]). Hence eachnon-finitely generated nilpotent group cannot be embedded in a finitely generatednilpotent group. However every finitely generated nilpotent group embeds in some2-generated nilpotent group of sufficiently large class [8]. Similarly, every polycyclicgroup embeds in a 2-generated polycyclic group [9].Let G be a group and g, f ∈ G. Further in the paper g f denotes f − gf (conjugateof g by f ) and [ g, f ] stands for g − f − gf (commutator of g and f ). Also [ g, f ; 1]means [ g, f ] and inductively [ g, f ; k + 1] stands for [[ g, f ; k ] , f ] , k = 1 , , .... By G ′ wedenote the derived subgroup of G . Then G ab = G/G ′ is the abelianization of G. Z means the infinite cyclic group and Z n denotes a cyclic group of order n. Recall that the Cartesian wreath product of groups is defined as follows. Let A and B be groups and D a group of all functions f : B → A with multiplication( f f )( x ) = f ( x ) f ( x ) for x ∈ B. The group B acts on D from the left by shiftautomorphisms: f b ( x ) = f ( bx ) for all f ∈ D, b, x ∈ B, and the associated with thisaction semidirect product D ⋊ B is called the Cartesian wreath product of the groups A and B, denoted by AW rB.
The subgroup D is called base subgroup of A W r B .Thus, every element of
AW rB has a unique presentation as bf ( b ∈ B, f ∈ D ) and MBEDDING THEOREMS FOR SOLVABLE GROUPS 3 the multiplication rule follows from the conjugation formula(1.1) f b ( x ) = f ( bx )in AW rB for any b, x ∈ B and f ∈ D. If instead of D one takes the smaller groupconsisting of all functions with finite support, that is, functions taking only non-identity values on a finite set of points, then one obtains a subgroup of A W r B called the wreath product ( direct wreath product ); it is denoted by A wr B .2.
Main results
The following question was posed by V. H. Mikaelian and A. Yu. Olshanskii in[5] and was also written by A. Yu. Olshanskii in [4](Question 18.73):Does every finitely generated solvable group of derived length l ≥ l + 1? Or at least, into some k -generated( l + 1)-solvable group, where k = k ( l )?We prove the following embedding theorems that imply an affirmative answer toMikaelian-Olshanskii ′ question. In the following statements, M means an arbitraryvariety of groups and A is the variety of abelian groups. For s ∈ N , A s means thevariety of abelian groups of exponent s. Theorem 1.
Let G be a countable group such that the abelianization G ab is a freeabelian group. Then G embeds in some -generated subgroup H of the Cartesianwreath product GW r Z . Corollary 2. (1)
Let G ∈ M be a countable group such that the abelianization G ab is a free abelian group. Then G embeds in some -generated group H ∈ M A. In particular, every finitely generated group G ∈ M such thatthe abelianization G ab is torsion-free, embeds in some -generated group H ∈ M A. (2) Let G be a countable solvable group of derived length l such that the abelian-ization G ab is a free abelian group. Then G embeds in some -generatedsolvable group H of length l + 1 . In particular, every finitely generated solv-able group G of derived length l such that the abelianization G ab is torsion-free, embeds in some -generated solvable group H of length l + 1 . (3) Every finitely generated group G ∈ M has a subgroup K of finite index thatcan be embedded in some -generated group H ∈ M A. In particular, everyfinitely generated solvable group G of derived length l has a subgroup K offinite index that can be embedded in some -generated solvable group H oflength l + 1 . Theorem 3.
Let G be a countable group such that the abelianization G ab is adirect product of a free abelian group and a finite group. Then G embeds in some -generated subgroup H of the Cartesian wreath product G W r Z . Corollary 4. (1)
Let G ∈ M be a countable group such that the abelianization G ab is a direct product of a free abelian group and a finite group. Then G embeds in a -generated subgroup H ∈ M A. In particular, every finitelygenerated group G ∈ M embeds in some -generated group H ∈ M A. (2) Let G be a countable solvable group of derived length l such that the abelian-ization G ab is a direct product of a free abelian group and a finite group. VITALY ROMAN’KOV
Then G embeds in a -generated solvable group H of length l + 1 . In par-ticular, every finitely generated solvable group G of derived length l embedsin some -generated solvable group H of length l + 1 . Theorem 5.
Let G be a group generated by a finite set u , ..., u m of elements offinite orders l , ..., l m , respectively. Then G embeds in some -generated subgroup H of ˜ W s = G wr ( Z s × Z s ) where s = lcm( l m , ..., l mm ) . Corollary 6. (1)
Let G ∈ M be a m -generated group of exponent e . Then G embeds in a -generated group H ∈ MA s where s = e m +1 , and so hasexponent e m +2 . (2) Let G ∈ M be a finite group. Then G embeds in some -generated finitegroup H ∈ M A . In particular, every finite solvable ( p - ) group G of derivedlength l embeds in some -generated finite solvable ( p - ) group H of length l + 1 . Proof of Theorem 1.
Let G be a countable group such that ¯ A = G ab is a free abelian group with basis { ¯ a i : i ∈ I ⊆ N } . Denote by a i a preimage of ¯ a i in G and set A = gp( a i : i ∈ I ).Let C be an infinite cyclic group generated by c and U = G W r C ≃ G W r Z . We are to show that G embeds in some 2-generator subgroup H of U , and thusprove Theorem 1. Now D U denotes the base group of U. Let s < s < ... < s i < ... be a sequence of positive integers such that s i + j − s j = s k + l − s l if and only if i = k and j = l. For definiteness, we take s i = 2 i for i = 1 , .... This sequence is called a strictly uneven sparse sequence .For brevity, denote c i = c s i , i = 1 , , ..., and set C = { c i : i = 1 , , ... } . Suppose that G is generated by a set of generating elements { g j : j ∈ J ⊂ N } . Let g j = a ( j ) g ′ j , where a ( j ) ∈ A, g ′ j ∈ G ′ , j ∈ J. Then G = gp( a i , g ′ j : i ∈ I, j ∈ J ).Let(3.1) g ′ j = r j Y q =1 [ u j,q , v j,q ]; u j,q , v j,q ∈ G ; j ∈ J. Let H = gp( c, d ) where d ∈ D U is defined as follows. Let C ⊆ C , C = A ⊔ U ⊔V ( |A| = | I | , |U| = |V| = |{ u j,q : q = 1 , ..., r j , j ∈ J }| = |{ v j,q : q = 1 , ..., r j , j ∈ J }| )is a disjoint union of sets. Here I is in one-to-one correspondence ι with indexes ofelements in A , the set of all elements of the form u j,q is in one-to-one correspondence δ with the set of indexes of all elements in U , and the set of all elements of the form v j,q is in one-to-one correspondence λ with the set of indexes of all elements in V .Then we set d ( c ι ( i ) ) = a i for each i ∈ I ; d ( c δ ( u j,q ) ) = u j,q , for each u j,q ;(3.2) d ( c λ ( v j,q ) ) = v j,q for each v j,q , and d ( c s ) = 1 in all other cases.First, we prove that H contains all the elements ˜ g ′ j ∈ D U such that ˜ g ′ j (1) = g ′ j and ˜ g ′ j ( c s ) = 1 for s = 0. For any pair ( j, q ) , we compute by direct computationthat(3.3) [ d c δ ( uj,q ) , d c λ ( vj,q ) ](1) = [ u j,q , v j,q ] . MBEDDING THEOREMS FOR SOLVABLE GROUPS 5
We set ˜ g ′ j = [ d c δ ( uj,q ) , d c λ ( vj,q ) ] . It remains to verify that every value ˜ g ′ j ( c s ) istrivial for each s = 0 . This statement follows because the sequence s , s , ..., isstrictly uneven sparse, and therefore each other non-trivial value d c κ ( uj,q ) ( c s ) , s = 0 , meets the trivial value of d c λ ( vj,q ) ( c s ). Then we get, by (3.1), that gp(˜ g ′ j : j ∈ J )(1)= G ′ , therefore gp(˜ g ′ j : j ∈ J ) ≃ G ′ . Obviously, ˜ g ′ j ∈ H for every j ∈ J. Secondly, d c ι ( i ) (1) = a i for each i ∈ I. Denote ˜ a i = d c ι ( i ) for i ∈ I . We set ˜ G =gp(˜ a i , ˜ g ′ j : i ∈ I, j ∈ J ). Then ˜ G (1) = G. We must show that ˜ G ≃ G (1), andconclude that ˜ G ≃ G. Obviously, there is a natural homomorphism µ : ˜ G → ˜ G (1) with the image ˜ G (1) . Obviously, the restriction of µ to gp(˜ g ′ j : j ∈ J ) is an isomorphism of ˜ G ′ onto G ′ . Suppose that z = z (˜ a i , ..., ˜ a i k , ˜ g ′ j , ..., ˜ g ′ j q ) ∈ ker ( µ ). Then the sum of exponents σ i t of ˜ a i t in z is 0 for each t = 1 , ..., k. Since the sequence s , s , ..., is strictly unevensparse all other nontrivial values of ˜ a i t corresponding to the its occurs in z are σ i t -exponents of the corresponding values of ˜ a i t , therefore are trivial. This valuesdon’t depend from other factors of z. Thus, z ∈ ˜ G ′ and therefore z = 1 . Hence µ isan isomorphism, and G embeds in H .Theorem is proved. Proof of Corollary 2.
Let G ab = ¯ A × ¯ T , where ¯ A is a free abelian group withbasis { ¯ a i : i ∈ I } , as before, and ¯ T is a finite abelian group. Let a i be a preimageof ¯ a i in G for i ∈ I. We define K = gp( a i , G ′ : i ∈ I ). This subgroup has a finiteindex in G. Then we define elements ˜ a i , ˜ g ′ j for i ∈ I and j ∈ J as above. We set˜ G = gp(˜ a i , ˜ g ′ j : i ∈ I, j ∈ J ). Then ˜ G (1) = K, and ˜ G ≃ ˜ G (1). Hence ˜ G ≃ K. This can be confirmed by the same argument as in the proof of Theorem 1.Corollary is proved. 4.
Proof of Theorem 3.
First, we prove a number of auxiliary statements.Let G be a group and V = G W r Z be the Cartesian wreath product of G andthe infinite cyclic group Z = gp( b ). Denote by D V the base group of V. For any u ∈ G , let u (0) ∈ D V be the constant function u (0) : B → G, u (0) ( b i ) = u, i ∈ Z . Then [ u (0) , b ] = 1 . Lemma 7.
For any element d ∈ D V there is an element x ∈ D V for which (4.1) d = [ x, b ] . Moreover, for any u ∈ G there is a unique x for which x (1) = u. Proof.
Let d = ( ... d − , d − , d , d , d , ... ) and x = ( ... x − , x − , x , x , x , ... ) , where d i = d ( b i ) and x i = x ( b i ) . Then (4.1) is equivalent to the system of equations(4.2) x − j x j +1 = d j , j ∈ Z . After setting x = u, u ∈ U, we uniquely compute for i ≥
1, that(4.3) x i = ud ...d i − and x − i = ud − − ...d − − i . (cid:3) In other words, d is a discrete (right) derivative of x , and x is a discrete integralof d . This integral is uniquely defined by d and its value x (1) = u . We will denoteit as I ( d, u ) and write I ( d, u ) ′ = d. Then we define I ( d, u ) = I ( I ( d, u ) , u ) ..., and so on. For simplicity, we keep u for all integrals. VITALY ROMAN’KOV
Corollary 8.
For each u ∈ G , there are a series of elements u ( k ) ∈ D V , k = 1 , , ..., for which (4.4) [ u ( k ) , b ] = u ( k − and u ( k ) (1) = u. In particular, (4.5) [ u ( k ) , b ; k ] = u (0) and [ u ( i ) , b ; k ] = 1 for i < k. Proof.
We define u ( k ) = I k ( u (0) , u ) for k = 1 , , ... . (cid:3) Now, let G be a group and W = G W r Z be a Cartesian wreath product, where Z = gp( b ) × gp( b ) is the free abelian group of rank 2 with basis { b , b } . Let D W denote the base group of W consisting of all functions f : Z → G, f ( i, j ) = f ( b i , b j ) = u ij , i, j ∈ Z . For any u ∈ G we consider the subgroup D u = gp( u ) W r Z of W generated by b , b and all functions f : Z → gp( u ) which make up the base group D u ≤ D W .Let f ( i, j ) = f ( b i , b j ) = u α ij , α ij ∈ Z , i, j ∈ Z . Then for any i ∈ Z , the set C ( i )( K ) of elements of the form K = { u α i ,j , j ∈ Z } , will be called i - column of D u , and for any j ∈ Z the set R ( j )( L ) of elements of the form L = { u α i,j , i ∈ Z } will be called j - row of D u . The set K can be considered as element of the basegroup D ,u of V ,u = gp( u ) W r gp( b ), and similarly the set L can be treated aselement of the base group D ,u of V ,u = gp( u ) W r gp( b ).Let C ( i )( w (0) ) , be a constant column corresponding to w ∈ gp( u ). By Corollary8, we get a series of columns C ( i )( w ( t ) ) such that C ( i )( w ( t ) ) ′ = C ( i )( w ( t − )and C ( i )( w ( t ) )( i ,
1) = w, t = 1 , , ... . Then [ C ( i )( w ( t ) ) , b ; t ] = C ( i )( w (0) ) and[ C ( i )( w ( t ) ) , b ; t + r ] = 1 for every r ≥ . Similarly, we get the elements R ( j )( w ( t ) ) , t = 0 , , , ... , that satisfy the follow-ing properties: [ R ( j )( w ( t ) ) , b ; t ] = R ( j )( w (0) ) and [ R ( j )( w ( t ) ) , b ; t + r ] = 1 forevery r ≥ . Now we are ready to prove a key lemma that allows us to distinguish individualelements of a given finite set, while at the same time making other elements of thisset trivial. We are dealing with the group W = G W r (gp( b ) × gp( b )) definedabove. Lemma 9.
Let u , ..., u t be a finite set of nontrivial elements of G . Let u (0) i ∈ D W be a constant function with the value u i . Then there exist functions f i ∈ D W , allof whose values belong to gp( u i ), which satisfy the following properties. (4.6) [ f i , b ; i ; b , t − i ] = u (0) i , [ f j , b ; i ; b , t − i ] = 1 for i = j, i, j ∈ { , ..., t } . Proof.
To construct f i we define its 0-row R (0)( u ( t − i ) i ) = ( ...u α − , i , u α , i , u α , i , ... ) . Then we build each column as C ( j )( u α j, i ) ( i ) and we have as a result f i . By construction every j th column of [ f i , b ; i ] is aconstant function with value u α ,j i , j ∈ Z . Then[ f i , b ; i, b ; t − i ] = u (0) i and [ f q , b ; i, b ; t − i ] = 1 for q < i. It happens because the columns are constructed as discrete integrals.
MBEDDING THEOREMS FOR SOLVABLE GROUPS 7
Let q > i.
By construction R ( u ( t − q ) q , b ; t − i ] = 1 . In other words, this is truefor a 0-row that does not change during the process of differentiating columns.Consider a more general case. Assume that u ∈ G and q ≥
0. For r ≥
0, we fix0-row R (0)( u ( r ) = ( ...u α , − , u α , , u α , , ... ) . . Then we expand this row to element f ∈ D V by adding the columns C ( j )( u α j, ) ( s ) for some s ≥
0. Obviously, for s = 0we have [ f, b ; r + 1] = 1 . We will prove by induction on s that this equality is truein general case.Let it is true for s −
1. The value u α ij of the function [ f, b ; r +1] in any point ( i, j )can be computed as follows. There is a Z -linear function L ( α i,j , α i +1 ,j , ..., α i + r +1 ,j .This function does not depend from s . By our assumptions, the value of thisfunction for any s is 0 for j = 0 and any i. By the assumption of induction, for s − j .Then this is true for j = 1 and j = −
1. Indeed, if the 1-row for s is( ..., u β − , , u β , , u β , , ... )and 0-row is ( ...u γ − , , u γ , , u γ , ... ) , then 0-row for s − ..., u β − , − γ − , , u β , − γ , , u β , − γ , , ... ) . Then by the assumption of induction L ( β i,j − γ i,j ) = L ( β i,j ) − L ( γ i,j ) = − L ( γ i,j = 0 . Similarly, this can be proved for j = −
1. Continuing, we will get that this is truefor s and each j . (cid:3) We proceed directly to the proof of the Theorem 3.Let G be a countable group such that the abelianization G ab is a direct productof a free abelian group ¯ A with a basis { ¯ a i : i ∈ I ⊆ N } and a finite abelian group¯ U = gp(¯ u , ..., ¯ u t ). Let a i denote a preimage of ¯ a i and u j denote a preimage of ¯ u j in G . Let A = gp( a i : i ∈ I ) and U = gp( u , ..., u t ).Consider the Cartesian wreath product ˜ W = G W r ( C × B ), where C = gp( c )is an infinite cyclic group and B is a free abelian group with base { b , b } . Then˜ W ≃ G W r Z . By D ˜ W we denote the base group of the group ˜ W .
First, we will do the same as in the proof of Theorem 1. Let s < ... < s i < ... be a strictly uneven sparse sequence of positive integers, i.e., s i + j − s j = s k + l − s l if and only if i = k and j = l. For definiteness, we take s i = 2 i for i = 1 , , .... Forbrevity, denote c i = c s i , i = 1 , , ..., and set C = { c i : i = 1 , , ... } . Suppose that G is generated by a set of elements { g m : m ∈ M ⊂ N } such that G ′ = gp( g m,m ′ = [ g m , g ′ m ] : m, m ′ ∈ M ). Then G = gp( a i , u j , g mm ′ : for i ∈ I, j ∈{ , ..., t } , m, m ′ ∈ M ).Let H = gp( c, b , b , d ) where d ∈ D ˜ W is defined as follows.Let C ⊆ C be a disjoint union { c , ..., c t } ⊔ I ⊔ M where |I| = | I | , |M| = | M | .Here I is in one-to-one correspondence ι with the set of indexes of elements in I , M is in one-to-one correspondence µ with the set of indexes of elements in M .Let D W be the base group in W = G W r gp(( b ) × gp) b ) and u (0) ∈ D W denote a constant function with the value u ∈ G. Let f ( j ) ∈ D W , j = 1 , ..., t, VITALY ROMAN’KOV are the elements constructed in Lemma 9. When constructing d we use f ( j ) , j =1 , ..., t ; a (0) i , i ∈ I, and g (0) m , m ∈ M. All values of d belong to D W .Then we set d ( c j ) = f ( j ) , j = 1 , ..., t ;(4.7) d ( c ι ( i ) ) = a (0) i , i ∈ I ; d ( c µ ( m ) ) = g (0) m , m ∈ M. and we set d ( c s ) = 1 in all other cases when s
6∈ C . Then G (0) = gp( u (0) j , a (0) i , g (0) m : j = 1 , ..., t ; i ∈ I, m ∈ M ) ≃ G. For any u (0) , one has u (0) , b ] = u (0) , b ] = 1 . We note also that G (0) = gp( u (0) j , a (0) i , g (0) m,m ′ : j =1 , ..., t ; i ∈ I, m, m ′ ∈ M )For any h ∈ D, h (1) means h (1 , , h (1) when all other values h ( c i , b j , b k ) are trivial.At first we prove by direct computation that H contains all elements of the form˜ g (0) m,m ′ (1) (remind that c i means c s i ) :(4.8) ˜ g (0) m,m ′ (1) = [ d c µ ( m ) , d c µ ( m ′ ) ] , m, m ′ ∈ M. Similarly we get(4.9) ˜ u (0) j (1) = [ d, b ; j, b , t − j ] c j , j = 1 , , ..., t. If | I | ≥
2, we cannot get ˜ a (0) i in the similar way. Instead we will use the followingelements:(4.10) ¯ a (0) i (1) = ( d c ι ( i ) , i ∈ I ) . We have(4.11) ˜ G = gp (˜ u (0) j (1) , ˜ g (0) m,m ′ (1) , a (0) i (1) : j = 1 , ..., t ; m, m ′ ∈ M, i ∈ I ) ≃ G. Let(4.12) ¯ G = gp(˜ u (0) j (1) , ˜ g (0) m,m ′ (1) , ¯ a (0) i (1) : j = 1 , ..., t ; m, m ′ ∈ M, i ∈ I ) ≤ H. There is a natural homomorphism (projection) ν of ¯ G onto ˜ G. In fact, ν is anisomorphism. Indeed, suppose that for some word z we have(4.13) z (¯ a (0)1 (1) , ..., ¯ a (0) k (1) , ˜ u (0)1 (1) , ..., ˜ u (0) t (1) , ˜ g (0) m .m ′ (1) , ..., ˜ g (0) m q ,m ′ q (1)) ∈ ker( µ ) . Since a , ..., a k induce a part of base of the free abelian group G ab every exponentsum σ i of ˜ a (0) i (1) , i = 1 , ..., k, in z is 0 . Then every other value corresponding toentries of ˜ a i in z is trivial. Then g is independent of ¯ a i (1) for each i ∈ I. Therefore, g has only trivial values outside of 1. It follows that ¯ G ≃ G. Theorem is proved.
Remark 10.
If the group G is finitely generated, then the proof of Theorem 3 canbe carried out without introducing elements of the form a i , i ∈ I. MBEDDING THEOREMS FOR SOLVABLE GROUPS 9 Proof of Theorem 5
Let us see what can be said if a group G is generated by a finite set of elementsof finite orders.Let G be a group and V = G W r Z be a wreath product of G and Z = gp( b ).As usual D V means the base group of V . Lemma 11.
Let u ∈ G be an element of a finite order l. Suppose, that everycomponent of d ∈ D V belongs to gp( u ) and d is a non-constant periodic function.This means that there is a number r > (period) such that d ( b i + r ) = d ( b i ) for all i. Let d (1) = u i , i ∈ N . Let x ∈ D V satisfy the condition [ x, b ] = d and x (1) = u i . Then x is periodicwith period lr .Proof. Let d = ( ...d − , d − , d , d , d , ... ) and x = ( ...x − , x − , x , x , x , ... ) , where d i = d ( b i ) and x i = x ( b i ) . We find x as in the proof of Lemma 7.Then for any i ≥
0, by (4.3), x i + lr = lr − Y j =0 u i d j = u i o and x − i − lr = − lr Y j = − u i d − j = u i , because, the product of all elements from l periods is 1. Therefore x ( i + r ) = x ( i )for all i. (cid:3) Corollary 12.
Suppose that the conditions of Lemma 11 are satisfied. Let V lr = G wr Z lr be the wreath product of G with the cyclic group Z lr = gp( b lr ) of order lr . By D V lr we denote the base subgroup of V lr . Since d and x are lr -periodic theycan be considered as elements of D V lr . Then d = [ x, b lr ] in V lr . Let u ∈ G, u = 1 and u l = 1 . Then u (1) = ( ...u − , u − , u , u , u , ... )is obviously l -periodic. By Lemma 11, u (2) is l -periodic, and so on.We can consider the elements u (0) , u (1) , ..., u ( t ) as elements of D V lt , the basesubgroup of V l t = G W r Z l t , where Z l t = gp( b l t ) is the cyclic group of order l t . It follows that we have the following finite analogue of Corollary 8.
Corollary 13.
For each u ∈ G, u l = 1 , there are a series of elements u ( k ) ∈ D V lt , k = 1 , , ..., t for which (5.1) [ u ( k ) , b ] = u ( k − and u ( k ) (1) = u. In particular, (5.2) [ u ( k ) , b ; k ] = u (0) and [ u ( i ) , b ; k ] = 1 for i < k. Lemma 14.
Let u , ..., u m ∈ G be elements of finite orders l , ..., l m , respectively.Let s = lcm( l m , ..., l mm ) . Let V s = G wr Z s be the wreath product of G with thecyclic group Z s = gp( b s ) of order s . By D V s we denote the base subgroup of V s .Then each of the elements u ( i ) i , i = 1 , ..., m that can be constructed by Corollary 13can be considered as element of D V s . These elements have the following properties: (5.3) [ u ( k ) i , b s ] = u ( k − i , [ u ( k ) i , b s ; k ] = u (0) i for k = 1 , ..., m and (5.4) [ u ( i ) , b s ; k ] = 1 for i < k. The following lemma is an analogue of Lemma 9.
Lemma 15.
Let u , ..., u m ∈ G be elements of finite orders l , ..., l m , respectively.Let s = lcm( l m , ..., l mm ) . Let W s = G wr Z s be the wreath product of G with the directproduct gp( b ,s ) × gp( b ,s ) of two cyclic groups of order s each. Let u (0) i ∈ D W s bea constant function with the value u i . Then there exist functions f i ∈ D W s , all ofwhose values belong to gp( u i ) , which satisfy the following properties. (5.5) [ f i , b ,s ; i ; b ,s ; t − i ] = u (0) i , [ f j , b ,s ; i ; b ,s ; t − i ] = 1 for i = j, i, j ∈ { , ..., m } . The proof completely repeats the proof of Lemma 9, taking into account Lemma14.We proceed directly to the proof of Theorem 5. We keep the notation introducedabove.Now G is a group generated by a finite set u , ..., u m of elements of finite orders l , ..., l m , respectively. We can assume that m ≥ . By Lemma 15, we construct thewreath product W s = G wr Z s and elements f i ∈ D V s , i = 1 , ..., m, which satisfythe equalities (5.5).Let ˜ W s = G wr ( Z s × Z s ) , where Z s = gp( c s ) and Z s = gp( b s ) × gp( b s ). Let s < s < ... < s m be a strictly uneven sparse sequence of positive integers suchthat s i + j − s j = s k + l − s l if and only if i = k and j = l. For definiteness, we take s i = 2 i for i = 1 , ...m. Since s ≥ m this property is valid modulo s ≥ m +1 .The rest of the proof completely repeats the arguments of the proof of Theorem3. Theorem is proved.Proof of Corollary 6. Now we can take in the proof of Theorem 5, instead of theactive group Z s × Z s in ˜ W s , the group Z e m +1 × Z e , and get the statement 1). Thestatement 2) follows directly from 1).Corollary is proved. References [1] Ph. Hall. Finiteness conditions for solvable groups. Proc. London Math. Soc. 4 (1954),pp. 419–436.[2] Ph. Hall. Nilpotent groups. Canad. Math. Cong. Summer Sem., University of Alberta,1957, 12–30.[3] G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups. J.London Math. Soc. 24 (1949), pp. 247–254.[4] The Kourovka Notebook. Unsolved problems in group theory. (Editors E. I. Khukhroand V. D. Mazurov). 19, (Russian Academy of Sciences. Siberian Branch. SobolevInstitute of Mathematics, Novosibirsk, Russia, 2018).[5] V. H. Mikaelian, A. Yu. Olshanskii. On abelian subgroups of finitely generatedmetabelian groups. J. Group Theory. 16 (2013), pp. 695–705.[6] B.H. Neumann, H. Neumann. Embedding theorems for groups. J. London Math. Soc.34 (1959), pp. 465-479.[7] A. Yu. Olshanskii. On Kaluzhnin-Krasner’s embedding of groups. Algebra DiscreteMath. 19 (2015), pp. 77–86.[8] V. A. Roman’kov. Embedding theorems for nilpotent groups. Siberian Math. Journal,13 (1972), no. 4, pp. 859–867.[9] V. A. Roman’kov. An embedding theorem for polycyclic groups. Math. Notes, 14(1973), no. 5, pp. 983–984.
MBEDDING THEOREMS FOR SOLVABLE GROUPS 11 [10] V. A. Roman’kov. Essays in group theory and cryptology: solvable groups. (Dosto-evsky Omsk State Univ. Publisher, 2017).
Additional information