Kovács, L. G.

Neumann, Bernhard Hermann

English

Kovács L. G.

Neumann Bernhard Hermann

University of Szeged

An embedding theorem for some countable groups By L. G. KOVÁCS and B. H. N E U M A N N in Canberra (Australia) Every countable soluble group can be embedded in a soluble 2-generator group, the solubility length increasing by no more than 2 in the process: this was shown in [5]. We here extend this result to some of the transfinite generalizations of soluble groups. The method of [5] has to be modified to do this, firstly as in [4] and secondly as in H A L L ' S paper [1]. . -We use the following notation and terminology. An ascending series of sub-groups of a group G is a family {L;}0S}£<T of subgroups of G indexed by the set of ordinals less than or equal to the ordinal o, and such that L0 = {1} and, for 0 < /. ̂  a <0 , - • [This condition ensures that I ^ S l , whenever juSA, and simultaneously that Lr is the union of its predecessors when A is a limit ordinal.] If each Lx is normal in its successor L> + i , or even in G, the series is called "normal" or "invariant", respect-ively. If for OSX [£; .+ 1 , J L ^ - J S L ; . , o r e v e n [G, + where [A, B] stands for the mutual commutator group of A and B, then the series is called "soluble" or "central", respectively. A soluble series is necessarily normal, and a central series invariant. If G has a soluble series with La=G, then G is defined to be an SN*-group; if the soluble series can be chosen invariant, then G is an SI* -group; if G has a •central series with L„=G, then G is a Z/i-group. Thc ordinal a is called a "length" of G — we do not assume it chosen minimal, and if G has 5Ar*-length or Sf*-length or ZJ-length a, then it has also every greater length. We shall prove the following theorem. , T h e o r e m . Every countable SI*-group G of length a can be embedded in a 2-generator SI* -group of length a + 2. The method of proof yields rather more than the theorem. To every countable group G, we contract a 2-generator group H which embeds it. The new feature of H is. that its second derived group is contained in a certain interdirect power Na •of G. Let © be a class of groups which is closed under the operations of taking sub-groups and taking interdirect powers like N„. (The reader has to refer to the first paragraph of the proof: an interdirect power F is selected there, and N„ is a restricted 140 L. G. Kovacs and B. H. Neumann direct power of F.) It follows from our construction that every countable group in © can be embedded in a 2-generator group whose second derived group is in fi. S o m e examples of classes which satisfy the conditions on £ are those of SN* ^groups, Z/f-groups, locally nilpotent groups, locally finite groups, periodic groups, etc. In p a r t i c u l a r , it f o l l o w s t h a t every countable SN*-group of length a is embeddable in a 2-generator SN*-group of length a+ 2. We mention an easy consequence of the theorem itself: C o r o l l a r y . There exist SI*-groups that are not locally soluble. This fact was pointed out by HALL in [2]; in the present context it follows by applying the theorem to a countable insoluble SI* -group G, for instance to one of the characteristically simple groups of MCLAIN [3]. P r o o f of the T h e o r e m . In addition to the notation introduced above, we also use the definitions and notation of [5]. In the complete wreath product P = G Wr C of the given SY*-group G and an infinite cyclic group C generated by an element c, we single out a subgroup that contains the restricted wreath product G wr C, but is not much larger. In the base group of P, that is the cartesian power Gc consist-ing of all functions on C to G, we single out those func t ions / that are constant for all sufficiently large positive powers of c, and also for all sufficiently large negative powers of c, the constant in this latter case being 1; thus we consider those / to which there is an integer p S 0, depending on f such that f(c") = I when / ( c " ) = / ( c " + 1 ) when n>p. These functions form a subgroup Fof Gc, and Fis normalized by C. We put FC = P°. The cartesian powers Lck are normal subgroups of Gc, but they will not in general form an ascending series in Gc, as the analogue of (I) may fail for limit ordinals ?.. However, if we put, for O^AScr, M, = / P. / . - , . so that M} consists of those functions f£'F that take values in L, , then each W/A is a normal subgroup of M„ = F and indeed of P°, and in fact {M^}^^ is an ascending soluble invariant series of P°. We omit the easy verification. If we put Ma+l =P°, then the thus-augmented series shows that P° is an SI* -group of length a+\. Next we take an infinite cyclic group B generated by an element b and form the complete wreath product Q=P° Wrfi . This contains in its base group P0B the direct powers Nx of the that is the functions on B to Mk with finite support. These are easily seen to form an ascending soluble invariant series {7VA}0SJS(,+ 1 in Q. We now use the assumption that G is countable, and generate it by a family {£•};?/ of elements indexed by the set / of positive integers. To these we define elements € F by k-,(c") = 1 when n<0, kt{c") = g f ! when n S 0. An embedding theorem for some countable groups 141 Put gn ={kl, c]; then g/ iC) g f ( C ) = ' l when n ^ 0. Thus the family {gn};si generates the coordinate subgroup (7, of Gc; clearly Cj £s G. Next we define an element a € .pPB by a(b) — c, a(b2') = k, when ; '£/. a(b") = 1 when n is. not a power of 2. . Let H be.the subgroup of Q generated by a and b, and let A be the normal closure of . a in H. Ther. A is generated by the conjugates say, of a, where n ranges over all integers. We now show that the derived group A' of A is contained in Na. First we remark that A' is generated by all commutators [«,„, a0] a r )d their conjugates under powers of b; and as b normalizes Na, it suffices to show that every [am, a0] lies in Na. Now [a,„, a0] is a function on B to P°, and we compute its value at b": [am, a0](b") = [ajb"), a0{b")} = [a(b"-"% a(b")}; this is 1 unless n — m and n are distinct powers of 2, say n—m= 2', n=2J, with i,j non-negative integers. In this case m = 2J — 2', and to any one m there is at most one such pair i , j . Thus the support of [a,„, o0] consists of at most one element of B; it only remains to show that the one non-trivial value of [am,a0], if it has one at all, lies in Ma = F. Now if m = 2J — 2' ^ 0, then [am,a0](b2J) = [a(b2'),a(b2J)]= g j i if ; = 0, = gn if y = 0, ; = [A :„k j ] if These values all lie in F, and it follows that A'^N„ as claimed. Incidentally the above argument also shows how G can be embedded i n . / / ; for if we put, for / £ I, hi = [ai-2i,ci0], • then bi(b)=gn, hi(b") = \ w h e n « ^ 1; hence the subgroup of H generated by {/*,•},is isomorphic to and thus to G. Finally we put Kk = HH Nx for O s ' ^ o , Then, as {N?}0SXStT is an ascending soluble invariant series of Q, also {Kx}0S1S(I is an ascending soluble invariant series of H. Adding Ka+1 =A and Ka+2 =H to this series, we obtain an ascending soluble invariant series that terminates with H itself; for as we have just seen, A'^N„ and thus also K'a+1 and obviously also H'^A. It follows that H is an SI* -group of length. <7 + 2, and the Theorem is proved. 142 L. G. Kovács, and B. H. Neumann: An embedding theorem for some countable groups References [1] P. HALL, The Frattini subgroups of finitely generated groups, Proc. London Math. Soc., (3) 11 (1961), 327-352 . [2] P. HALL, On non-strictly simple groups, Proc. Cambridge Philos. Soc., 59 (1963), 531—553. [3] D . H . MCLAIN, A characteristically simple group, Proc. Cambridge Philos. Soc., 5 0 ( 1 9 5 4 ) , 641-642 . [4] B . H . N E U M A N N , Embedding theorems for ordered groups, J. London Math. Soc., 3 5 ( 1 9 6 0 ) , 503-512 . [5] B . H . NEUMANN and HANNA NEUMANN , Embedding theorems for groups, J. London Math. Soc., 34 (1959), 4 6 5 - 4 7 9 . THE AUSTRALIAN NATIONAL UNIVERSITY (Received April 10, 1964) 

An embedding theorem for some countable groups

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An embedding theorem for some countable groups

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