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On Dimension Subgroups and the Lower Central Series . Abstract ON DIMENSION SUBGROUPS AND THE LOWER CENTRAL SERIES . ABSTRACT AUTHOR: Graciela P. de Schmidt TITLE OF THESIS: On Dimension Subgroups and the Lower Central Series. DEPARTMENT Mathematics • DEGREE: Master of Science. SUMMARY: The aim of this thesis is to give an ex- position of several papers in which the relationship between the series of dimension subgroups of a group G and the lower central series of G is studies. The first theorem on the subject, proved by W. Magnus,asserts that when G is free the two series coincide term by terme This thesis presents a proof of that theorem together with the proofs of several related results which have been obtained more recently; moreover, it gathers together the various techniques from combinatorial group theory, commutator calculus, group rings and the the ory of graded Lie algebras, which have been used in studying this topic. ON DIMENSION SUBGROUPS AND THE LOWER CENTRAL SERIES by Graciela P. de Schmidt A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH IN PARTIAL FULFILMENT OF THE REQUlREMENTS FOR THE DEGREE OF MASTER OF SCIENCE Department of Mathematics McGill University Montreal May 1970 ® Graciela P. de Schmidt (i) PREFACE The aim of this thesis is to give an exposition of several papers in which the relationship between the series of dimension subgroups of a group G and the lower central series of G is studied. It has been conjectured that for an arbitrary group the two series coincide term by terme W. Magnus [12] proved that this is indeed the case for free groups and several authors have generalized his result in various directions. The conjecture has now been proved for sorne special classes of groups [9], [11]; however, up to now, the attack on the problem for an arbitrary group has only yielded the identity of the first three terms of the series [7]. Several approaches have been taken in dealing with this problem: the techniques of combinatorial group theory, commutator calculus, group rings and the theory of graded Lie algebras have been mixed in various proportions. In order to elucidate and unify our exposition of sorne of the main theorems on the subject appearing in the literature, it is necessary to present a lot of background material relevant to these topics. In conclusion l would like to thank Professor W. Jonsson who introduced me to this problem through a seminar and has been of great assistance at all times. (ii) TABLE OF CONTENTS page §1. CENTRAL SERIES 1 §2. POLYCYCLIC GROUPS 12 §3. THE COLLECTING PROCESS 19 §4. GROUP RING AND DIMENSION SUBGROUPS 32 ~ §5. THE POINCARE-BIRKHOFF-WITT THEO REM 42 §6. MAGNUS THEO REM 51 §7. CASES n = 2, n = 3 FOR AN ARBITRARY GROUP 62 § 8. SOME FUR'l~HER RESULTS 72 REFERENCES 77 §l. CENTRAL SERIES In this section we state and prove sorne results on central series and nilpotence, of which we will make use later, they can be found for instance as part of a more complete exposition in the notes of Philip Hall 12] or in the book on Theory of Groups by Marshall Hall [1]. The notation is standard in works on Group Theory but we recall sorne for the more frequently used concepts. For any group G, 1 will denote the identity element of Gand also the subgroup of G consisting of the identity alone. If x,y E G, then [x,y] will denote the commutator x-ly-lxy, and for n > 2 one defines recursively the simple commutator [xl'···,xn] = [Ixl, ••• ,xn_l],xn]. If H,K are subgroups of G, then [H,K] will denote the subgroup of G generated by '{[x,y], x E H, Y E K}, and similarly we define [Hl, ••• ,Hn] for subgroups Hl, ••• ,Hn of G. In general, we denote by <x. , i E I> the group ~ generated by the set {xi' i El}. (1.1) Lemma. Let G be any group, x,y,z E G. Then z (i) [x,yz] = Ix, z] [x,y] , (ii) [xy, z] = [x,z]y[y,z]; (iii) Ix,y -1 ,z] Y Iy,z -1 ,x] z Iz,x-1 ,y] x = 1; 2. Pro'of: xY = x[x,y] and xy = yxlx,y] = yxY• So yz x = x[x,yz], but a1so Cance11ation of x in both expressions of xYz yie1ds (i). Next, (xy)z = xy[xy,z], and a1so cance11ation of xy gives (ii). Now let u = xzx -1yx, and v,w be obtained from u by cyc1ic permutations of x,y,z. Then [x,y-1,z]y = y-1[x,y-1]-lz-l[x,y-1]zy = = y-1y(x-1y-1xz-1x-1 ) (yxy-1zy) = u -1v. -1 z -1 -1 x -1 Simi1ar1y [y,z ,x] = v w, [z,x ,y] = w u and (iii) ho1ds. Since -1 Y -1 -1 -1 -1 -1 -1 -1 [x,y ,z] = x y x(yy }z (yy )x yxy zy = = [x,y] (z-l)Y[y,x]zy = [[y,x],zY] we obtain (iv) by rep1acing this and the corresponding expressions for the other two factors in eLi-i) 3. D'ef'initions. Let Gi be a subgroup of G, i = 0, l , • •• • An asc'e'ndingser'ies is a series (1.2) A descending series is a series (1.2') > l . These concepts of course coincide for f'inite series (1.3) (1.4) A descending series is called a C'en'tral series if G. ~ is a normal subgroup of G for i = 1,2, ••• and Gi/Gi +l is in the center of G/Gi +l , or equivalently if [G,Gi ] ~ Gi +l , i = 1,2, •••• Similarly one defines ascending central series. Instances of central series are: (1.5) the upper central series ~ G , where Zi+1G is defined to be the subgroup of G such that Zi+1G/ZiG is the center of G/ZiG, and ZOG = 1. Thus ZlG is the center of G. (1.6) the lower centra'l' se'r'i'es G = r l G ~ r 2G ~ ••• -> l 4. where one defines rlG = Gand ri+1G = [G,riG] for i = 1,2,.... r 2G = [G,G] = G' is called the derived group or the commutator subgroup of G. The subgroups r.G,1. i = 1,2, ••• are not only normal subgroups of G, but also fully invariant, since any group homornorphism maps commutators into commutators. These series are called "upper" and "lower" because of the following (1.7) Lemma. (a) For any descending central series such as (1.2') r.G < G. j = 1,2, ••• ,. ) J (b) for any ascending central series (1.2) Z.G1. -> G.1. i = 0,1,2, •••• Proof: We prove these relations by induction on i and j. For i = 0, j = 1 they hold by definition. Let G. > r.G for sorne j ~ 1; then ) - ) since the series Gl ~ G2 > is central. This proves (a). G~ for sorne then For (b), assume Z1.,G -> 1. i ~ 0; gG. onto the map of G!G.1. into G;ZiG that sends the coset 1. 5. the coset gZiG, g e: G, is onto with kernel ZiG/Gi. Now Gi+l/Gi is in the centre of G/Gi by definition of a central series, so its image G'+lz.G/z.G must lie in the center of ~ ~ ~ G/ZiG, i.e. in Zi+lG/ZiG. Hence and the proof is complete. (1.8) Definition. A group G is called nilpotent if there exists an integer k > 1 such that rkG = l, i.e. if the lower central series is fini te. We say that a fini te series (1.3) has leng:th r, r > l, if 1 but 1. The c'l'ass of a nilpotent - Gr + l = Gr =1 group G is by definition the length c of the lower central series of G, if G =1 1 and it is 0 if (and only if) G = 1. Thus Abelian groups are precisely the nilpotent g~oups of class 1. (1.9) Lemma. If K is a normal subgroup of G, then r, (G/K) = Kr, (G)/K. In particular G/K is nilpotent J J if and.. only if K contains sorne term of the lower cenlttral series. Proof: If 7T: G -+ G/K is the canonical projection, then since rjG is fully invariant 6. Since ~ is onto, equality holds. (1.10) Lemma. If G is nilpotent of class c, then every subgroup and every factor group of G is also nilpotent, of class at most c. (1.11) Lemma. If IGI = pn > l, where p is a prime, then G is nilpotent of class at most n. pr'oof: The center of a fini te p-group is bigger than the identity alone (cf. [1], theorem 4.3.1), so ZIG> 1; by the same reason ZiG> Zi_lG, i ~ l, since G/Zi_lG is also a finite p-group. One sees by induction on i that IG : ZiGI ~ pn-i, so the upper central series terminates and ZnG = G. Since the lower central series is contained in it, we must have rn+lG = 1. (1.12) Lemma. Let G be the group generated by a set A, let B be a subset of G, and let H = <BG> where BG =' {b,g " g,E G b E B} • Then [H,G] = <cG>, where c = {[a,b]; a E A, b E B}. G Proof: Let K = <C >. Clearly H and K are normal in G and so K -< [H ,G] • Now consider G/K; since [a,b] E K for aIl a E A, b E B by definition of K, every generator a of G' commutes with every b E B mod K, so BK/K is contained in the center of G/K, and the same is true for HK/K.
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