Rapporto di Ricerca CS G. Busetto, E. Jabara

UNIVERSITÀ CA FOSCARI DI VENEZIA Dipartimento di Informatica Technical Report Series in Computer Science Rapporto di Ricerca CS-2005-12 Ottobre 2005 G. Busetto, E. Jabara Some observations on factorized groups Dipartimento di Informatica, Università Ca Foscari di Venezia Via Torino 155, 30172 Mestre Venezia, Italy

Some observations on factorized groups. Giorgio Busetto - Enrico Jabara Abstract. We consider a factorized group G = AB where A and B are subgroups of G. We introduce the concept of (A, B, X)-orbit of an element of G and we study the relation between the (A, B, X)-orbit of an element and the element itself; our main result relates the cardinality of the (A, B, X)- orbit of an element of G to the order of the element. 1. Introduction. Let G be a group, and A, B subgroups of G. The group G is the product of its subgroups A and B whenever G = AB, namely G = { ab a A, b B }; sometimes we say that G is factorized by A and B with the same meaning. A and B are usually called the factors of G. Writing G as a product of two subgroups is usually called a factorization of G. The factorization is proper if both the factors are proper subgroups of G; a group is said to be factorizable if it admits a proper factorization. Factorizable groups occur very often in group theory and the literature on this subject is wide. Generally speaking, in most cases, the approach to the study of factorizable groups can be described, roughly, as follows: given certain properties of the factors, how these properties affect the structure of the whole group? The first papers specifically devoted to the study of factorized groups appear around 1940; since then a lot has been done, and in 1992 a monograph has been published, containing a large part of the theory developed until then [1]. However, although many good results have been obtained, in our opinion still some major sides of this theory remain unexplored, and it is partly still obscure. We think that these obscure parts need to be clarified if people want really satisfactory results. We would like to spend a few words in the attempt to justify and explain why we hold this opinion. We begin recalling two well known classical results on 1

factorized groups: one is due to Itô in 1955 [6], and it states that any group (finite or infinite) which is the product of two abelian subgroups, is metabelian. The other, known as Wielandt-Kegel s theorem, concerns only with finite groups, and states that a group which is the product of two finite nilpotent subgroups is soluble ([7], [13]); its proof was completed in 1961. Unfortunately Wielandt-Kegel s theorem does not say anything if there is any connection between the nilpotency classes of the factors and the derived length of the whole group. However, in the light of Itô s theorem, the following famous conjecture took shape: Conjecture. Any group which is the product of two nilpotent subgroups is soluble and its derived length is bounded by a function of the nilpotency classes of the factors. Moreover, it was also hypothized that this function is just the sum of the nilpotency classes, so that Itô s theorem would be just a particular case. This conjecture was proved in some special cases; for example for finite groups where the factors have coprime orders ([5], assuming G soluble, since Wielandt-Kegel s theorem has been proved later); this result is also an immediate corollary of the following result ([8]): if G = AB is finite and A and B are nilpotent of classes respectively α and β, then the (α + β)-th term of the derived series of G is a group such that any prime dividing its order must divide both the orders of A and of B. Also, it is known that, in the same hypotheses as before, the (α + β 1)-th term of the derived series of G is nilpotent ([4]). On the other side no considerable progress has been obtained when the group is finite and nilpotent, and this makes hard to work when you leave the hypothesis of finiteness of the group. Hence, also in the light of moving to infinite factorized groups, one of the main problems to consider is to prove or disprove the conjecture in the case of finite p-groups, where p is a prime. However, recently, Cossey and Stonehewer ([3]) exibit some examples of finite factorized groups with nilpotent factors, where the derived length of the group is 1 plus the sum of the nilpotency classes of the factors, so disproving the strongest form of the conjecture. Nevertheless the general opinion is that the conjecture still holds, with a slightly different function. One could ask why we insist so much on the property of solubility of a factorized group in connection with the nilpotency of the factors and why this particular feature is connected with what we have called an obscure part of the theory of factorized groups. First of all, Kegel [7] proved that a finite group is solvable if and only if G = G 1 G 2... G n where G i is nilpotent and 2

G i G j = G j G i for i, j {1, 2,... n}. This result is no longer true in general for infinite groups, as a counterexample constructed by Marconi shows (see [1], proposition 7.6.3). In any case, when you move from finite to infinite groups, the investigation appears far more difficult, due, in our opinion, to the lack of tools for this kind of investigation. In the infinite case some kind of pathological features can happen. For example, an obvious consideration is that, if G = AB is finite, then G = A B A B, and so the primes dividing the orders of the elements of G are the same as those dividing the orders of the elements of A or B; in other words no new prime can show up. On the other hand, in the infinite case, there exist examples where the behaviour is very different, and pathological in some sense; in 1982 Sysak ([11]) found an example of a periodic locally soluble group G = AB where A and B are isomorphic p-groups, where p is a prime number, and G contains elements of finite order coprime to p; actually this is a particular case of Sysak s example, but we do not want to get in in more details here. In 1984-85 Suchkov ([9], [10]) exibited a countable group G = AB, where the factors are locally finite and G contains a free subgroup of countably infinite rank and also contains a periodic subgroup which is not locally finite. On the other side if we assume as hypotesis that the group is soluble, this things cannot happen; for, if G = AB is soluble and the factors are periodic, no new prime or element of infinite order can show up in G ([11]); actually it is sufficient that G satisfies a condition weaker then solubility, but again we do not want to get into more details here. We only observe that the hypothesis of local solubility is not sufficient, by Sysak s example, to avoid pathological cases; hence the weakening of the condition of solubility must be of a different kind. According with the relevance of the property of solubility in the study of factorized groups, we observe that the great majority of positive results in this area in the infinite case is obtained under the additional hypothesis that the factorized group is soluble or it is close to be soluble. With the help of this hypothesis the group has a sufficiently good structure to be able to work on it. What it is not clear, here it is the obscure part of this theory, is when and how a factorized group behaves well, namely has a reasonably good structure, depending only on the structure and properties of the factors, without the help of precautional hypotheses on the whole group. Of course the case when the factors are abelian is an exception to our argument since, by Itô s theorem quoted above, in this case the factorized group is metabelian, and this means that its structure is good enough to work on such a group. So we could say that the theory of factorized groups with abelian factors is satisfactory. Even if there are some other Itô s like results, essentially the case with abelian factors is the only treatable case in 3

the way we would like to be for infinite groups. Almost always, if the factors are not abelian, the only known way to proceed is to impose solubility s like properties on the whole group, eluding in this way the major difficulties. Until now, in this introduction, we have made an attempt to give some motivations to our opinion that there is still a lot to do in this area if we want to obtain a good theory. Proving or disproving the conjecture mentioned above would be an important step, but how to proceed? We do not know the answer; however we feel that we must have a better knowledge of the way in which the factors of a factorized group permute; we shall define more precisely the meaning of way of permuting in the next section. This kind of investigation has not been so much developed. We recall papers by Zappa ([14],[15]) and by Szep ([12]) but in our opinion there is still a lot to discover. The present paper brings a small contribution to the knowledge of the way of permuting of the factors, introducing the concept of (A, B, X)-orbit of the elements of a factorized group G = AB. 2. The (A, B, X)-orbit of an element of G. In the sequel, unless explicitly stated, G will always be a group factorized by its subgroups A and B, namely G = AB and we shall denote by X the intersection of A and B, namely X = A B. We have G = G 1 = (AB) 1 = B 1 A 1 = BA, since A and B are closed with respect to take inverses. Then, if g G, there are elements a 1, a 2 A and b 1, b 2 B such that g = a 1 b 1 = b 2 a 2. If a 1, a 2 are elements of A and b 1, b 2 are elements of B such that b 1a 1 = b 1 a 1 = a 2b 2 = a 2 b 2, then x 1 = a 1 a 1 1 = (b 1 ) 1 b 1 and x 2 = (a 2 ) 1 a 1 2 = b 2 b 2 are elements of X, and a 1 = x 1a 1, b 1 = b 1x 1, a 2 = a 2x 1 2, b 2 = x 2b 2. Conversely, let x 1, x 2 X, and let a 1 = x 1a 1 A, b 1 = b 1x 1 1 B, a 2 = a 2x 1 2 A, b 2 = x 2b 2 B. Then we can write the element g as g = b 1 a 1 = a 2 b 2 and also as g = b 1 a 1 = a 2 b 2. In particular, b 1, a 1 and a 2, b 2 are uniquely determined by the element g if and only if X = 1. Remark 1. If X = 1, G induces in a natural way a bijection A B ϕ A B, defined by (a, b) ϕ = (a 1, b 1 ) if ba = a 1 b 1. We could say that ϕ describes how A and B permute. ϕ induces on the set G = A B a group structure in the following way: for a, a A, b, b B, define (a, b) (a, b ) = (aa 1, b 1b ) where (a 1, b 1) = (a, b) ϕ. In 1940 Zappa([14], reported also in [15]), shows that G endowed with this operation is a group, and the map 4

G Φ G (ab) (a, b) a A b B is an isomorphism; in particular G = AB, where A = { (a, 1) a A } = A and B = { (1, b) b B } = B. Actually we must say that Zappa uses a slightly different notation; our notation is more convenient for our specific aim. Conversely if a bijection A B ϕ A B is given is it possible to decide when ϕ is induced by a factorized group G = AB with A B = 1? In this case this is equivalent to induce a group structure on the set G in the way described above. Zappa([14]) gives necessary and sufficient conditions for ϕ to induce a group structure on G. Unfortunately, in our opinion, the sufficient conditions given by Zappa are very theoretical and very difficult, if not impossible, to be checked in practice. Moreover, Zappa s method allows to construct groups of the form AB, where A = A, B = B and A, B are given groups, but only with A B = 1 (see also Casadio [2]). Our first aim is to extend partly the above considerations for any X; in order to do this we introduce an equivalence relation X in the set A B by defining, if (a, b), (a, b ) A B, then (a, b) X (a, b ) if and only if ab and a b are conjugate by an element of X; denote by A B/ X the factor set induced by X, and by (a, b) the equivalence class containing (a, b). It is easily seen that the relation X can be defined also in this way: if (a, b), (a, b ) A B, then (a, b) X (a, b ) if and only if there exist x, y X such that a = xay 1 and b = ybx 1. Note that the relation X does not depend on the particular factorized group that we are considering, but depends only on A, B and X. We shall make use of the following: Lemma 2.1 Let a 1, a 2 A and b 1, b 2 B. Then a 1 b 1 and a 2 b 2 are conjugate by an element of X if and only if b 1 a 1 and b 2 a 2 are conjugate by an element of X. Proof. Suppose that exist x X such that (a 1 b 1 ) x = a 2 b 2. Then a 1 2 ax 1 = b 2b x 1 = z X and we have a 2 = a x 1 z 1 and b 2 = zb x 1. Therefore b 2 a 2 = (b 1 a 1 ) xz 1 with xz 1 X. The proof of the converse is analogous. For any X, in analogy to the case X = 1 described in Remark 1, we have: 5

Proposition 2.2 The position (a, b) ϕ = (a 1, b 1 ) if ba = a 1 b 1 in G gives ϕ rise to a well defined bijection of A B/ X A B/ X. We will call ϕ the bijection of A B/ X induced by G. Proof. If (a, b) = (a, b ), ab and a b are conjugate by an element of X. Therefore, by Lemma 2.1 ba and b a are also conjugate by an element of X. Hence (a, b) ϕ = (a, b ) ϕ, namely ϕ is well defined. The proof of the injectivity is almost the same as the proof of the well definition; surjectivity is clear. In analogy with Remark 1, ϕ describes how A and B permute. We would like to characterize these bijections, in a way that can be really used. We do not go so far in the present paper, however we make a little step forward; in Section 3, we give a numerical necessary condition that ϕ has to satisfy if it is induced by a factorized group. In order to do this, we are going to define the (A, B, X)-orbit of an element g G. We define inductively a sequence of subsets Γ i (g) of G, i Z, as follows: put Γ 0 (g) = g X. Then, suppose that Γ i (g) is defined for i 0; choose an element h Γ i (g); write h = ba for some a A, b B, and let Γ i+1 (g) = (ab) X For i 0, again assume that Γ i (g) is defined; choose an element h Γ i (g); write h = ab for some a A, b B, and let Γ i 1 (g) = (ba) X. Remark 2. Note that the choice of the element h in Γ i (g) and of the elements a A, b B is not unique in general. We show that the sequence, Γ i (g), i Z, depends only on the choice of g and not on the choice of these other elements. Of course Γ 0 (g) depends only on the choice of g. Thus, suppose that Γ i (g) depends only on the choice of g for some i 0 and choose an element h Γ i (g), h = ba for some a A, b B. Then, by definition of Γ i (g), ba and ba are conjugate by an element of X. Hence, by Lemma 2.1, ab and ab are also conjugate by an element of X, and this implies that (ab) X = (ab) X, so that Γ i+1 (g) depends only on the choice of g. The argument is analogous for i 0. Definition 2.3 We will call the sequence:..., Γ n (g),..., Γ 1 (g), Γ 0 (g), Γ 1 (g),..., Γ n (g),... ( ) 6

the (A, B, X)-orbit of g. Moreover, we will call the cardinality of the underlying set { Γ i (g) i Z } the length of the (A, B, X)-orbit of g. Remark 3. By definition, two elements g and h of G have the same (A, B, X)-orbit if and only if Γ i (g) = Γ i (h), i Z. Looking at the way in which it is constructed, the sequence ( ) is determined if it is known the i-th term of the sequence ( ) for one fixed i Z. The 0-th term of the sequence determines which elements of G have that particular sequence as their (A, B, X)-orbit, and they form a subset of G of the form g X for some g G. In Proposition 2.4 we collect some elementary observations about the (A, B, X)-orbits of the elements of G. Proposition 2.4 The following hold: i) (i) Let g G and i Z; then Γ i+1 (g) = { ab a A, b B, ba Γ i (g) } and Γ i 1 (g) = { ba a A, b B, ab Γ i (g) }. ii) If h Γ i (g), then Γ j (h) = Γ i+j (g) i, j Z. iii) g G and i, j Z, either Γ i (g) = Γ j (g) or Γ i (g) Γ j (g) =. iv) the set of subsets of G: { i Z Γ i(g) g G } form a partition of G. v) i Z Γ i(g) g A g B for all g G. vi) If g G, then Γ i (g) = 1 i Z if and only if X C G (g). Thus the (A, B, X)-orbit of every element of G reduces to a sequence of elements of G, if and only if X Z(G). In particular this happens when X = 1 or A and B are both abelian. Proof. (i) and (ii) follow easily from Lemma 2.1. (iii): the conjugation by X is an equivalence relation. (iv): the relation defined in G by putting h and k in relation if and only if i, j Z, g G such that h Γ i (g) and k Γ j (g) is an equivalence relation. (v): if h Γ i (g), k Γ i+1 (g), then h and k are conjugate by an element of A and also by an element of B; for, we have Γ i (g) = h X, where h = ba for some a A, b B; then Γ i+1 (g) = (ab) X ; we have ab = (ba) a 1 = (ba) b ; k is conjugate to ab by an element of X = A B and therefore h and k are conjugate by an element of A and also by an element of B; almost the same argument holds for k Γ i 1 (g). Then (v) follows easily. (vi) is a simple consequence of the definition of (A, B, X)-orbit. 7

Observe that if A is normal in G, X = 1 and g = ab, where a A and b B (uniquely determined by g in this case), then the (A, B, X)-orbit of g describes the conjugates of a under the action of b. More precisely, in this particular case, the (A, B, X)-orbit of g is the following sequence:..., a b n b,..., a b 1 b, ab,..., a bn b,.... 3. The length of the (A, B, X)-orbit of an element of G. We are going to observe that there is a connection between the order of an element g G and the length of its (A, B, X)-orbit. We begin by: Proposition 3.1 If the length of the (A, B, X)-orbit of g is a finite number n, then the length of the (A, B, X)-orbit of g n is equal to 1. Proof. Let g = b 0 a 0, a 0 A, b 0 B. Write b 0 a 0 = a 1 b 1,..., b i a i = a i+1 b i+1 for i = 1, 2,..., n and a i A, b j B. Note that b j a j Γ j (g). We obtain g n = (a 1 b 1 ) n = a 1 a 2... a n b n... b 2 b 1 and b n... b 2 b 1 a 1 a 2... a n = (b n a n ) n. By hypothesis, b n a n Γ 0 (g). Hence (b n a n ) n = ((a 1 b 1 ) y ) n = (a 1... a n b n... b n ) y for some y X. Therefore the length of the (A, B, X)- orbit of g n is equal to 1, as required. Then we have: Theorem 3.2 Let I g = { m Z Γ m (g) = Γ 0 (g) }. Then: a) I g is an ideal of Z. b) I g {0} if and only if the (A, B, X)-orbit of g has finite length l and, in this case, I g = l. c) I g = J g := { m Z g m = ab, a A, b B and Γ 0 (g a ) = Γ 0 (g) }. Proof. (a) Let r, s I g. Then, by Proposition 2.4 (ii) and (iii) we have Γ r (g) = Γ s (g) if and only if Γ r s (g) = Γ s s (g) = Γ 0 (g); so r s I g. (b) by Propositon 2.4 (ii), the (A, B, X)-orbit of g has finite length l if and only if Γ l (g) = Γ 0 (g) and l is the smallest positive integer having this property. (c) Let g = b 0 a 0, a 0 A, b 0 B and m Z, m 0. Write b 0 a 0 = a 1 b 1,..., b i a i = a i+1 b i+1 for 0 i m and a i b i = b i 1 a i 1 for 0 i m +1, where a i A, b i B. Note that b i a i Γ i (g). Thus we obtain 8

g m = a 1 b 1... a 1 b }{{} 1 = a 1 a 2... a m b m... b 2 b 1. m Let a = a 1 a 2... a m, b = b m... b 2 b 1, so that g m = ab. Then If m < 0, let n = m. Hence g a = (a 1 b 1 ) a 1...a m = a m +1 b m +1 = b m a m. g m = (g 1 ) n = a 1 0 b 1 0... a 1 0 b 1 0 }{{} n = a 1 0... a 1 n+1 b 1 n+1... b 1 0. Let a = a 1 0... a 1 n+1, b = b 1 n+1... b 1 0 so that g m = ab. Then (g 1 ) a = (a 1 0 b 1 0 )a 1 0...a 1 n+1 = a 1 n b 1 n = a 1 m b 1 m = (b m a m ) 1, thus g a = b m a m. Hence g a = b m a m whenever m Z. If m I g, then b m a m Γ 0 (g); therefore Γ 0 (g) = Γ 0 (g a ); hence m J g. Conversely, if m J g, then Γ 0 (g a ) = Γ 0 (g) and therefore b m a m Γ 0 (g). But b m a m Γ m (g), that is to say that Γ m (g) = Γ 0 (g), and so, by (a), m J g. Note that, if X Z(G), and so the (A, B, X)-orbits consist of sequences of elements of G, Theorem 3.2 (c) says that the length of the (A, B, X)-orbit of g, if it is finite, is the smallest natural number n such that, if g n = ab, a A, b B, then g and a commute; this is, of course equivalent to say that g and b commute, since ab certainly commutes with g. Moreover, if g commutes with a, then, since ab is a power of g, a commutes with ab and therefore commutes with b. Corollary 3.3 The length of the (A, B, X)-orbit of g is equal to g : h, where h = ab a A, b B, ab g, Γ 0 (g a ) = Γ 0 (g). Proof. It follows immediately from Theorem 3.2 (a) and (c) We obtain a necessary condition for the bijection ϕ induced by G on the set A B/ X as described in Proposition 2.2. Theorem 3.4 Let be A B/ X A B/ X the bijection induced by G on A B/ X. If g G has finite order n, and g = ab, a A, b B, ϕ 9

then (a, b) ϕn = (a, b). In particular, if G has finite exponent m, then ϕ has finite order dividing m. Proof. The length of the (A, B, X)-orbit of g divides n by Corollary 3.3. Hence Γ n (g) = Γ 0 (g). If (a, b) ϕn = (a, b ), a A, b B, then, by definition of ϕ, a b Γ n (g); therefore a b Γ 0 (g), namely ab and a b are conjugate by an element of X. Thus (a, b ) = (a, b). The last part of the statement follows immediately. REFERENCES [1] Amberg, B. - Franciosi, S. and De Giovanni, F. Products of groups. Oxford Mathematical Monographs, Oxford, 1992. [2] Casadio, G. Costruzione di gruppi come prodotto di sottogruppi permutabili. Rend. Mat. e Appl. 2 (1941), 348-360. [3] Cossey, J. and Stonehewer, S. E. On the derived length of finite dinilpotent groups. Bull. London Math. Soc. 30 (1998), 247-250. [4] Gross, F. Finite groups which are the product of two nilpotent subgroups. Bull. Australian Math. Soc. 9 (1973), 267-274. [5] Hall, P. and Higman, G. The p-length of a p-soluble group and reduction theorems for Burnside s problem. Proc. London Math. Soc. (3) 7 (1956), 1-42. [6] Itô, N. Über das Produkt von zwei abelschen Gruppen. Math. Z. 62 (1955), 400-401. [7] Kegel, O.H. Produkte nilpotenter Gruppen. Arch. Math. (Basel) 12 (1961), 90-93. [8] Pennington, E. On products of finite nilpotent groups. Math. Z. 134 (1973), 81-83. [9] Suchkov, N.M. An example of a mixed group factorized by two periodic subgroups. Algebra i Logika 23 (1984), 573-577 (Algebra and Logic 23 (1984), 385-387). [10] Suchkov, N.M. Subgroups of a product of locally finite groups. Algebra i Logika 24 (1985), 408-413 (Algebra and Logic 24 (1985), 265-268). [11] Sysak, Y.P. Some examples of factorized groups and their relation to ring theory. Infinite groups 1994 (Ravello), 257-269. de Grutyer, Berlin, 1996. 10

[12] Szep, J. Zur Theorie der faktorisierbaren Gruppen. Acta Sci. Math. (Szeged) 16 (1955), 54-57. [13] Wielandt, H. Über Produkte von nilpotenten Gruppen. Illinois J. Math. 2 (1958), 611-618. [14] Zappa, G. Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro. Atti del secondo congresso dell Unione Matematica Italiana (1940), 119-125 Cremonese, Roma, 1942. [15] Zappa, G. Fondamenti di Teoria dei Gruppi. Monografie Matematiche del C.N.R., vol. II, Cremonese, Roma, 1970. Girogio Busetto Dipartimento di Informatica Universitá di Ca Foscari Via Torino 155, I-30174 Mestre - Venezia, Italy e-mail: busetto@dsi.unive.it Enrico Jabara Dipartimento di Matematica Applicata Universitá di Ca Foscari Dorsoduro 3825/e, I-30122 Venezia, Italy e-mail: jabara@unive.it 11