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Groups with All Subgroups Subnormal

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Groups with All Subgroups Subnormal Groups with all subgroups subnormal Carlo Casolo Dipartimento di Matematica “Ulisse Dini”, Universit`adi Firenze, I-50134 Firenze Italy e-mail: [email protected]fi.it Otranto, 4 - 8 giugno 2007 2 Contents 1 Locally nilpotent groups 5 1.1 Commutators and related subgroups . 5 1.2 Subnormal subgroups and generalizations . 11 1.3 Classes of groups . 15 1.4 Nilpotent groups and their generalizations . 23 1.5 Classes of locally nilpotent groups . 29 1.6 Preliminaries on N1 ......................... 39 2 Torsion-free Groups 43 2.1 Locally nilpotent torsion–free groups . 43 2.2 Isolators . 45 2.3 Torsion–free N1-groups . 49 3 Groups of Heineken and Mohamed 55 3.1 General remarks . 55 3.2 Basic construction . 57 3.3 Developements . 62 3.4 Minimal non-N groups . 67 4 Bounded defects 71 4.1 n-Baer groups . 71 4.2 Roseblade’s Theorem . 74 4.3 First applications . 81 5 Periodic N1-groups 83 5.1 N1-groups of finite exponent . 83 5.2 Extensions by groups of finite exponent . 97 5.3 Periodic hypercentral N1-groups . 101 6 The structure of N1-groups 105 6.1 Solubility of N1-groups . 105 6.2 Fitting Groups . 107 6.3 Hypercentral and Smith’s groups . 112 6.4 The structure of periodic N1-groups . 119 3 4 CONTENTS 7 Beyond N1 125 7.1 Generalizing subnormality . 125 7.2 Groups with many subnormal subgroups . 129 7.3 The subnormal intersection property. 137 7.4 Other classes of locally nilpotent groups . 139 Chapter 1 Locally nilpotent groups In this chapter we review part of the basic theory of locally nilpotent groups. This will mainly serve to fix the notations and recall some definitions, together with some important results whose proofs will not be included in these notes. Also, we hope to provide some motivation for the study of groups with all subgroups subnormal (for short N1-groups) by setting them into a wider frame. In fact, we will perhaps include more material then what strictly needed to understand N1-groups. Thus, the first sections of this chapter may be intended both as an unfaithful list of prerequisites and a quick reference: as such, most of the readers might well skip them. As said, we will not give those proofs that are too complicate or, conversely, may be found in any introductory text on groups which includes some infinite groups (e.g. [97] or [52], for nilpotent groups we may suggest, among many, [56]). For the theory of generalized nilpotent groups and that of subnormal subgroups, our standard references will be, respectively, D. Robinson’s classical monography [96] and the book by Lennox and Stonehewer [64]. In the last section we begin the study of N1-groups, starting with the first basic facts, which are not diffucult but are fundamental to understand the rest of these notes. 1.1 Commutators and related subgroups Let x, y be elements of a group G. As customary, we denote by xy = y−1xy the conjugate of x by y. The commutator of x and y is defined in the usual way as [x, y] = x−1y−1xy = x−1xy. Then, for n ∈ N, the iterated commutator [x,n y] is recursively defined as follows [x,0 y] = x, [x,1 y] = [x, y] and, for 1 ≤ i ∈ N, [x,i+1 y] = [[x,i y], y]. 5 6 CHAPTER 1. LOCALLY NILPOTENT GROUPS Similarly, if x1, x2, . xn are elements of G, the simple commutator of weight n is defined recursively by [x1, x2, . , xn] = [[x1, . , xn−1], xn]. We list some elementary but important facts of commutator manipulations. They all follow easily from the definitons, and can be found in any introductory text in group theory. Lemma 1.1 Let G be a group, and x, y, z ∈ G. Then (1) [x, y]−1 = [y, x]; (2) [xy, z] = [x, z]y[y, z] = [x, z][x, z, y][y, z]; (3) [x, yz] = [x, z][x, y]z = [x, z][x, y][x, y, z]; (4) (Hall-Witt identity) [x, y−1, z]y[y, z−1, x]z[z, x−1, y]x = 1. Lemma 1.2 Let G be a group, x, y ∈ G, n ∈ N, and suppose that [x, y, y] = 1; then [x, y]n = [x, yn]. If further [x, y, x] = 1, then n n n n (xy) = x y [y, x](2). If X is a subset of a group G then hXi denotes the subgroup generated by X. If U and V are non-empty subsets of the group G, we set [U, V ] = h[x, y] | x ∈ U, y ∈ V i and define inductively in the obvious way [U,n V ], for n ∈ N. Finally, if A ≤ G, and x ∈ G, we let, for all n ∈ N,[A,n x] = h[a,n x] | a ∈ Ai. If H ≤ G, HG denotes the largest normal subgroup of G contained in H, and HG the normal closure of H in G, i.e. the smallest normal subgroup of G containing H. Clearly, \ g G g HG = H and H = hH | g ∈ Gi. g∈G More generally, if X and Y are non-empty subsets of the group G, we denote by XY the subgroup hxy | x ∈ X, y ∈ Y i. The following are easy consequences of the definitions. Lemma 1.3 Let H and K be subgroups of a group. Then [H, K] E hH, Ki. Lemma 1.4 Let X, Y be subsets of the group G. Then [hXi, hY i] = [X, Y ]hXihY i. If N E G, then [N, hXi] = [N, X]. The next, very useful Lemma follows from the Hall-Witt identity. 1.1. COMMUTATORS AND RELATED SUBGROUPS 7 Lemma 1.5 (Three Subgroup Lemma). Let A, B, C be subgroups of the group G, and let N be a normal subgroup such that [A, B, C] and [B, C, A] are con- tained in N. Then also [C, A, B] is contained in N. The rules in Lemma 1.1, as well as others derived from those, may be ap- plied to get sorts of handy analogues for subgroups. For instance, if A, B, C are subgroups of G and [A, C] is a normal, then [AB, C] = [A, C][B, C]. More generally, we have Lemma 1.6 Let N, H1,...,Hn be subgroups of the group G, with N E G, and put Y = hH1,...,Hni. Then [N, Y ] = [N, H1] ... [N, Hn]. The same commutator notation we adopt for groups actions: let the group G act on the group A. For all g ∈ G and a ∈ A, we set [a, g] = a−1ag, and [A, G] = h[a, g] | a ∈ A, g ∈ Gi. With the obvius interpretations, the properties listed above for standard group commutators continue to hold. For a group G, the subgroup G0 = [G, G] is called the derived subgroup of G, and is the smallest normal subgroup N of G such that the quotient G/N is abelian. The terms G(d) (1 ≤ d ∈ N) of the derived series of G are the characteristic subgroups defined by G(1) = G0 and, inductively , G(n+1) = (G(n))0 = [G(n),G(n)] (the second derived subgroup G(2) is often denote by G00). The group G is soluble if there exists an n such that G(n) = 1; in such a case the smallest integer n for which this occurs is called the derived length of the soluble group G. Of course, subgroups and homomorphic images of a soluble group of derived length d are soluble with derived length at most d. A group is said to be perfect if it has no non-trivial abelian quotiens; thus, G is perfect if and only if G = G0. By means of commutators are also defined the terms γd(G) of the lower central series of a group G: set γ1(G) = G, and inductively, for d ≥ 1, γd+1(G) = [γd(G),G] = [G,d G]. These are also characteristic subgroups of G. A group G is nilpotent if, for some c ∈ N, γc+1(G) = 1. The nilpotency class (or, simply, the class) of a nilpotent group G is the smallest integer c such that γc+1(G) = 1. Lemma 1.7 Let G be a group, and m, n ∈ N \{0}. Then (1) [γn(G), γm(G)] ≤ γn+m(G);. (2) γm(γn(G)) ≤ γmn(G); From (1), and induction on n, we have 8 CHAPTER 1. LOCALLY NILPOTENT GROUPS (n) Corollary 1.8 For any group G and any 1 ≤ n ∈ N, G ≤ γ2n (G). In par- ticular a nilpotent group of class c has derived length at most [log2 c] + 1. Also, by using (1) and induction, one easily proves the first point of the following Lemma, while the second one follows by induction and use of the commutator identities of 1.1, Lemma 1.9 Let G be a group, and 1 ≤ n ∈ N. Then (1) γn(G) = h[g1, g2, . , gn] | gi ∈ G, i = 1, 2, . , ni. (2) If S is a generating set for G, then γn(G) is generated by the simple commutators of weight at least n in the elemets of S ∪ S−1. The upper central series of a group G is the series whose terms ζi(G) are defined in the familiar way: ζ1(G) = Z(G) = {x ∈ G | xg = gx ∀g ∈ G} is the centre of G, and for all n ≥ 2, ζn(G) is defined by ζn(G)/ζn−1(G) = Z(G/ζn−1(G)).
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