IA-automorphisms of groups with almost constant upper central series

Transcription

1 IA-automorphisms of groups with almost constant upper central series Marianna Bonanome, Margaret H. Dean, and Marcos Zyman Abstract. Let G be any group for which there is a least j such that Z j = Z j+1 in the upper central series. Define the group of j-central automorphisms as the kernel of the natural homomorphism from Aut(G) to Aut (G/Z j ). We offer sufficient conditions for IA(G) to have a useful direct product structure, and apply our results to certain finitely generated center-by-metabelian groups. 1. Introduction The IA-group of a group G, denoted by IA(G), consists of those automorphisms that induce the identity on G/G, where G is the commutator subgroup of G. The investigation of the IA-group has been of interest in different contexts. P. Hall, for example, has shown that the IA-group of a nilpotent group of class c is nilpotent of class c 1 (see [5]); and M. Zyman has remarked that if G is finitely generated nilpotent, so too is IA(G) (see [8]). A result of particular interest here is due to S. Bachmuth (see [1]), who has shown that the IA-group of a free metabelian group of rank two is equal to its inner automorphism group. He has also shown that this is not the case when the rank is larger than two. In this paper we study the IA-group of a group G for which the upper central series stalls at some point. This means that there exists a least positive integer j for which Z j = Z j+1. We refer to these groups as H j -groups, whose examination was inspired by an example of P. Hall in [6]. We discuss this example in 4. We define the group of j-central automorphisms of G, denoted by Aut cj (G), as the kernel of the natural homomorphism from Aut(G) to Aut (G/Z j ), where Z j is a term of the upper central series. In Lemma 3.1 we prove that for G an H j -group, the subgroup of Aut(G) generated by the inner automorphisms together with the j- central automorphisms is a direct product of these two groups, modulo Aut cj 1 (G). Let Aut zj (G) = IA(G) Aut cj (G). In Theorem 3.2, we give sufficient conditions 2000 Mathematics Subject Classification. 20F14, 20F16, 20F22, 20F28. Key words and phrases. Automorphism groups, center-by-metabelian groups, central automorphisms, H j -groups, IA-automorphisms, inner automorphisms, j-central automorphisms, solvable groups, upper central series. The third-named author was supported by The City University of New York PSC-CUNY research award program (grant # ). The second and third-named authors were supported by a BMCC faculty development grant. December 16,

2 2 MARIANNA BONANOME, MARGARET H. DEAN, AND MARCOS ZYMAN for the group of IA-automorphisms of an H j -group to equal the direct product of the inner automorphisms with Aut zj (G), modulo Aut zj 1 (G). The main result is: Theorem 3.3. Let G be a group for which there is a least j such that Z j = Z j+1 in the upper central series. If IA (G/Z j ) = Inn (G/Z j ), then IA(G)/Aut zj 1 (G) = ( Inn(G)Aut zj 1 (G)/Aut zj 1 (G) ) ( Aut zj (G)/Aut zj 1 (G) ). When Z 1 = Z 2, we get the very pleasing result: Corollary 3.4 Let G be a group such that Z 1 = Z 2 in the upper central series. If IA(G/ζ(G)) = Inn(G/ζ(G)), then IA(G) = Inn(G) Aut z1 (G). The paper is organized as follows: In 2 we provide definitions, notation, and a preliminary lemma; while 3 is devoted to the proof of our theorem. In 4 we consider some examples, motivated by the existence of a family of finitely generated metabelian groups whose IA-group is not finitely generated [2]. As further evidence of the complexity of solvable groups, we end 4 with the construction of continuously many finitely generated center-by-metabelian groups whose group of IA-automorphisms is not finitely generated. 2. Preliminary discussion We write G = gp(a) to express the fact that A is a generating set for G. We write a typical commutator as [g, h] = g 1 h 1 gh. For any group G define the group of IA-automorphisms (or IA-group) to be the kernel of the natural homomorphism from Aut(G) to Aut (G/G ). Thus, IA(G) = { α Aut(G) : g 1 (gα) G (g G) }. For any group G, Inn(G) IA(G) and G/ζ(G) = Inn(G). Throughout this paper, the inner automorphism induced by x G will be denoted as α x. For any positive integer j, we define the group of j-central automorphisms of G, denoted by Aut cj (G), to be the kernel of the natural homomorphism from Aut(G) to Aut (G/Z j ). A j-central automorphism acts as the identity on G modulo Z j. Thus: Aut cj (G) = { α Aut(G) : g 1 (gα) Z j (g G) }. We use the notation Aut c (G) = Aut c1 (G) and we refer to Aut c (G) as the group of central automorphisms of G. An alternate definition of Aut cj (G) is given by the following lemma: Lemma 2.1. For any group G, Aut cj (G) = { ϕ Aut(G) : [ϕ, α x ] Aut cj 1 (G) for all α x Inn(G) }.

3 IA-AUTOMORPHISMS OF GROUPS WITH ALMOST CONSTANT UCS 3 Proof. ϕ Aut cj (G) g 1 (gϕ) Z j for all g G x 1 ( ) xα g 1 (gϕ) = x 1 ( xαg 1 αg ϕ ) Zj 1 for all x, g G α 1 g α ϕ g = [α g, ϕ] Aut cj 1 (G) for all g G. As stated in 1, a group G is an H j -group if there exists a least positive integer j for which Z j = Z j+1. Since there is a natural correspondence between Aut zj (G) = Aut cj (G) IA(G) and G Z j, we can also characterize Aut zj (G) as Aut zj (G) = { α AutG : g 1 (gα) Z j G (g G) }. We remark that every nilpotent group of class c is trivially an H c -group, since G = Z c = Z c+1 =. A group G is termed center-by-metabelian if G/ζ(G) is metabelian. G is centerby-metabelian if and only if G ζ(g), where G = [G, G ]. The variety of center-by-metabelian groups contains interesting examples of H 1 -groups, some of which will be discussed in Proof of the main result Lemma 3.1. Let G be an H j -group. Then, modulo Aut cj 1 (G), gp ( Inn(G), Aut cj (G) ) = (Inn(G)Autcj 1 (G)) Aut cj (G). Proof. By Lemma 2.1, [Inn(G), Aut cj (G)] Aut cj 1 (G) for any group G. To prove that Inn(G) Aut cj (G) Aut cj 1 (G), suppose that α x Inn(G) Aut cj (G), where gα x = g[g, x]. Then for all g G, [g, x] Z j. This implies x Z j+1. Since Z j = Z j+1, we have x Z j, and thus [g, x] Z j 1 for all g G. Hence, Inn(G)Aut cj 1 (G) Aut cj (G) = 1 modulo Aut cj 1 (G), and our result follows. Note that if G is not an H j -group, then it is not necessarily the case that Inn(G) Aut cj (G) Aut cj 1 (G). For example, let G = a, t 0, t 1,... ; t 0 = 1, [a, t i ] = t i 1 (i = 1, 2,...), [t i, t j ] = 1. Observe that Z j = gp(t 1,..., t j ) and G has a strictly ascending upper central series. In particular G is not an H j -group for any j. Let T = gp(t 1, t 2,...). Any element of G can be written uniquely as g = a m t, for some m Z and some t T. Then, gα tj+1 = g[g, t j+1 ] = g[a m, t j+1 ]. Since [a, t j+1 ] = t j Z j, [a m, t j+1 ] Z j. Hence α tj+1 Aut cj (G). Moreover, since aα tj+1 = at j, then α tj+1 / Aut cj 1 (G). Our main result is: Theorem 3.2. Let G be a group for which there is a least j such that Z j = Z j+1 in the upper central series. If IA (G/Z j ) = Inn (G/Z j ), then, modulo Aut zj 1 (G), IA(G) = (Inn(G)Aut zj 1 (G)) Aut zj (G).

4 4 MARIANNA BONANOME, MARGARET H. DEAN, AND MARCOS ZYMAN Proof. It suffices to show that IA(G) = Inn(G)Aut zj (G). Then, by Lemma 2.1 and the proof of Lemma 3.1, IA(G) = Inn(G) Aut zj (G), modulo Aut zj 1 (G). Let ϕ IA(G). Then x 1 (xϕ) G G Z j for all x G. Thus, x 1 (xϕ)g Z j = G Z j for all x G. By hypothesis IA(G/Z j ) = Inn(G/Z j ), so there exists g G such that for all x G, (xϕ)z j = x g Z j. Hence, ( x g (xϕ) = x 1 (xϕ) g 1) g Zj, and consequently, x ( ) 1 xϕαg 1 Zj. I.e., ϕαg 1 Aut cj (G) IA(G) = Aut zj (G). Then ϕ = ( ) ϕαg 1 αg Aut zj (G)Inn(G). Corollary 3.3. Let G be a group such that Z 1 = Z 2 in the upper central series. If IA(G/ζ(G)) = Inn(G/ζ(G)), then IA(G) = Inn(G) Aut z1 (G). 4. Examples For the remainder of the paper, we will be considering H 1 -groups; for simplicity, we will refer to them as H-groups, and we will denote Aut cj (G) and Aut zj (G) as Aut c (G) and Aut z (G), respectively. Accordingly, The 1-central automorphisms will be simply termed central automorphisms. Although the group of IA-automorphisms of a two-generator metabelian group is metabelian [4], it need not be finitely generated. There is a family of examples of finitely generated metabelian groups whose group of IA-automorphisms is not finitely generated [2]. As further evidence of the complexity of solvable groups, we provide continuously many finitely generated center-by-metabelian groups whose group of IAautomorphisms is not finitely generated. The following results play a central role in the discussion of our examples. Lemma 4.1. Let G = gp(x 1,..., x n ) be a finitely generated H-group such that ζ(g) G. Suppose that there exists a presentation for G in which every relator is a product of commutators. Assume also that ζ(g) = gp(z 1, z 2,...). Then each map of the form ϕ ij : x i x i z j x k x k (k i) lifts to a central automorphism of G, and the set of all ϕ ij generates Aut c (G). Furthermore, if ζ(g) is freely generated by {z 1, z 2,...}, then Aut c (G) is freely generated by the ϕ ij. Proof. Let G = gp(x 1,..., x n ) be a finitely generated H-group, and let {z 1, z 2,...} be a generating set for ζ(g). Since G comes with a presentation in which every relator is a product of commutators, each ϕ ij extends to a map on G that sends defining relators to 1. Hence, each ϕ ij lifts to a homomorphism. Since G ζ(g), the inverse for ϕ ij can be constructed; therefore, ϕ ij is a central automorphism.

5 IA-AUTOMORPHISMS OF GROUPS WITH ALMOST CONSTANT UCS 5 Next we show that {ϕ ij } generates Aut c (G). Let ϕ Aut c (G) and write x i ϕ = x i m j=1 z αij j. Since central automorphisms act trivially on G and G ζ(g), x i n,m k,j=1 ϕ α kj kj = x i m j=1 ϕ αij ij = x i m j=1 z αij j = x i ϕ for each x i. It is easy to show that if the ϕ ij do not freely generate Aut c (G), then the z i do not freely generate ζ(g). This completes the proof. Lemma 4.2. Let G = gp(x 1,..., x n ) be a finitely generated H-group such that ζ(g) G. Suppose that there exists a presentation for G in which every relator is a product of commutators. If Aut c (G) is finitely generated, then ζ(g) is finitely generated. Proof. Suppose {α 1,..., α m } generates Aut c (G). Then x i α j = x i z ij with i = 1,..., n; j = 1,..., m; and z ij ζ(g). Suppose now that there exists a z ζ(g) such that z / gp(z 11,..., z nm ). Then we can define a central automorphism α z : x 1 x 1 z x i x i (i 1) which is not generated by {α 1,..., α m }, a contradiction. Example 1. Let G = F/[F, F ] be the free center-by-metabelian group of rank 2, where F = gp(x, y) is the free group on two generators. Then ζ(g) = G and G/ζ(G) is free metabelian of rank 2. It is well known that a free metabelian group has trivial center. Hence, G is an H-group, with Aut c (G) = Aut z (G). By a theorem of Bachmuth (see [1]), IA(G/ζ(G)) = Inn(G/ζ(G)), so we may apply Theorem 3.2 to conclude that IA(G) = Inn(G) Aut c (G). Ridley has shown in [7] that ζ(g) is free abelian of countably infinite rank. Thus, it follows from Lemmas 4.1 and 4.2 that Aut c (G) is itself free abelian of infinite rank. In particular, we have exhibited a finitely generated center-by-metabelian group whose group of IA-automorphisms is not finitely generated. Before proceeding, we state an important lemma. We are unaware of an existing proof in the literature. Lemma 4.3. If W = a t is the wreath product of an infinite cyclic group by another, then IA(W ) = Inn(W ). Proof. There is a standard expanded presentation for W that is convenient for us to use: W = a, t, a i (i Z); a 0 = a, a t i = a i+1, [a i, a j ] = 1 (i, j Z).

6 6 MARIANNA BONANOME, MARGARET H. DEAN, AND MARCOS ZYMAN By examining this presentation we deduce that W is free abelian, freely generated by { } [t, a] ti (i Z). It follows that W can also be regarded as a free module over Z[t, t 1 ], generated by the single element µ = [t, a]. The module structure allows us to show that every IA-automorphism is an inner automorphism. Let ϕ IA(G). Then ϕ : a aµ ra t tµ rt, where r a, r t Z[t, t 1 ]. Applying the commutator calculus and using the module structure of W we have: (i) [t, µ] = [t, [t, a]] = [t, a t a] = [t, a][t, a t ] = µµ t = µ 1 t, and hence: (ii) µϕ = [t, a]ϕ = [tµ rt, aµ ra ] = [t, a][t, µ ra ] = [t, µ] ra [t, a] = µ (1 t)ra+1. Similarly, if then µϕ 1 = µ (1 t)sa+1. Thus, Consequently, ϕ 1 : a aµ sa t tµ st, a(ϕ 1 ϕ) = (aµ sa )ϕ = aµ ra+[(1 t)ra+1]sa = a. r a + [(1 t)r a + 1]s a = 0. Therefore r a and s a are mutual divisors, so they are associates in Z[t, t 1 ]. As a result, (1 t)r a + 1 is a unit in Z[t, t 1 ]. The units of Z[t, t 1 ] are of the form ±t γ. If 1 + r a r a t = t γ, then r a (t 1) = t γ + 1. This implies that 1 is a root of t γ + 1, a contradiction. Hence 1 + r a r a t = t γ, and r a (1 t) = t γ 1. Thus, { (1 + t + + t r a = γ 1 ) if γ > 0, t t γ+1 + t γ if γ < 0. Finally, suppose r t = n 1 t i1 + n 2 t i2 + + n k t i k, where n j, i j Z. A straightforward computation shows that ϕ is the inner automorphism induced by w = t γ a n1 i 1 a n k i k W. Example 2. The following example is given by P. Hall in [6]. Let B be the free nilpotent group of class 2, freely generated by {b i : i Z}. Thus Consider the quotient group B =..., b 1, b 0, b 1,... ; [b i, b j, b k ] (i, j, k Z). N = B ; [b i, b j ] = [b i+1, b j+1 ] (i, j Z). For each positive integer r, set d r = [b 0, b r ]. In N, d r = [b i, b i+r ] for all i Z. In fact, the effect of the relations [b i, b j ] = [b i+1, b j+1 ] is that N is free abelian on {d 1, d 2,...}. Let a : b i b i+1

7 IA-AUTOMORPHISMS OF GROUPS WITH ALMOST CONSTANT UCS 7 be the translation map on the generators of N. By the nature of the defining relations of N, a induces an automorphism of N. Set b = b 0 and form the semidirect product G = N a = gp(b, a). It is straightforward to verify that ζ(g) = N. In particular, ζ(g) is a free abelian group of countably infinite rank, contained in G. As a consequence of the defining relations for G, it is the case that G/ζ(G) = Z Z. Since Z Z has trivial center, G is a two-generator group satisfying that ζ(g/ζ(g)) = 1. Thus, G is a two-generator H-group with Aut c (G) = Aut z (G), which is center-bymetabelian. G can also be viewed as a matrix group (see [3], page 45, for details). Since ζ(g) is not finitely generated, it follows from Lemma 4.2 that Aut c (G) is not finitely generated. By Lemma 4.3 and Theorem 3.2 we conclude that the IA-group of G is not finitely generated. Now, both Examples 1 and 2 lead to a collection of continuously many finitely generated center-by-metabelian groups whose IA-group is not finitely generated. Suppose G is either of the groups discussed in the above examples. Then ζ(g) is free abelian of countably infinite rank. Let X be a free generating set for ζ(g). It is known that there are continuously many subgroups G σ of ζ(g), each generated by a non-empty subset A σ of X, and that the quotient groups G/G σ fall into continuously many isomorphism classes ([3], page 3). For each G σ, ζ(g/g σ ) = {x G/G σ : [x, g] G σ (g G)}. Since G is an H-group and G σ ζ(g), we conclude that (4.1) ζ (G/G σ ) = ζ(g)/g σ. By (4.1), ζ(g/g σ ) is finitely generated only if the complement of A σ is finite. Discard the subsets of X whose complement is finite. A continuous family of subsets A σ X still remains, the corresponding quotient groups G/G σ fall into continuously many isomorphism classes, and each ζ(g/g σ ) is free abelian of countably infinite rank. It follows again from (4.1) that ζ (G/G σ ) (G/G σ ), and G/G σ / ζ (G/Gσ ) = G/G σ / ζ(g)/gσ = G/ζ(G). By Theorem 3.2 each G/G σ is a 2-generator center-by-metabelian group whose IA-group is not finitely generated. 5. Acknowledgements We would like to express our gratitude to Professor Peter Neumann for his helpful discussions. This work was inspired by a suggestion of his. We would also like to extend our thanks to Professor Gilbert Baumslag for his guidance and helpful comments; and to Professor Katalin Bencsáth for her generous encouragement. Finally, we thank the referee for offering a shorter version of the proof of Theorem 3.2, and for several other insightful comments.

8 8 MARIANNA BONANOME, MARGARET H. DEAN, AND MARCOS ZYMAN References 1. S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), S. Bachmuth, G. Baumslag, J. Dyer, and H. Y. Mochizuki, Automorphism groups of two generator metabelian groups, J. London Math. Soc. (2) 36 (1987) no. 3, K. Bencsáth, M. Bonanome, M. H. Dean and M. Zyman, Lectures on Finitely Generated Solvable Groups, soon to appear in the SpringerBriefs Series. 4. C. K. Gupta, IA-automorphisms of two generator metabelian groups, Arch. Math. 37 (1981), P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. 4 (1954), J. N. Ridley, The free centre-by-metabelian group of rank two, Proc. London Math. Soc. (3) 20 (1970), M. Zyman, IA-automorphisms and localization of nilpotent groups, Ph.D. Thesis-City University of New York, Department of Mathematics, The City University of New York-CityTech,, 300 Jay Street, Brooklyn, NY address: mbonanome@citytech.cuny.edu Department of Mathematics, The City University of New York-BMCC, 199 Chambers Street, New York, NY address: mdean@bmcc.cuny.edu Department of Mathematics, The City University of New York-BMCC, 199 Chambers Street, New York, NY address: mzyman@bmcc.cuny.edu