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
(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 =' {b,g " g,E G b E B} • Then [H,G] =
G Proof: Let K =
(1.13) Corollary. Let G = . Then rnG is generated by rn+lG together with aIl commutators [al, ••• ,an], ai E A.
Proof: Let B ={[al, ••• ,an-l]~ ai E A}, so H = rn_lG and rnG = [rn_lG,G] is generated by the commutators x u = [al, ••• ,an] and their conjugates. But u = u[u,x], so x u _ u mod rn+1G for aIl x E G.
(1.14) Lemma. Let A,B,C, be subgroups of G,
A* = [B,C,A], B* = [C,A,B] and C* = [A,B,C]. If N is a normal subgroup of Gand A*B* < N then C* < N.
Proof: Let x E A, Y E B, z E C. Then [y,z-l,x] and -1 [z,x ,y] belong to N and by (iii) of lemma (1.1) so does -1 [x,y ,z] since N is normal in G. Thus, every element z E C commutes mod N with the generators [x,y-l], hence, with aIl elements of [A,B], so that C* < N.
(1.15) Corollary. If A,B,C are aIl normal in G, then C* < A*B*.
(1.16) Theorem. Let K be a subgroup of G,
be a series of subgroups such that H, ~ is normal in H for i = 0,1,2, ••• , and [H" K] < H, +1' ~ - ~ , i = 0,1,2, ••• Let KI = K, KJ' = {x EK; [H, ,x] < H,+" ~ -- ~ J i = o,l, ••• }. Then
j,k = 1,2, ••• , 8. and
[!i.,r.K] < H.+. l. J - l.), aIl i and j.
Proof: By definition of K., j = 1,2, ••• , we have J and
[Hi,Kk,Kj ] < [Hi+k,Kj ] ~ Hi + j +k ; a1so Hk is normal in
[H.,[K.,K ]] < H·+·+ , l.) k - l.) k hence [KJ.,Kk] < K. k Th' th t K K K - J+. l.S proves a = 1 ~ 2 > ••• is a central series for K, so that r.K < K. and ) J [H. ,r .K] < H.+. l. J - l. J.
(1.17) Coro11ary. Let G be any group. Then
(i) [r.G,r.G] < r.+.G; l. J - l. J (ii) If i _> )', [Z. G Ir .G] < Z. .G; in particu1ar l. J l.-)
[z.G,r.G] = 1. l. l. Proof: (i) In theorem (1.16) take Hi = ri+1G and
K = G; then [H. 1,r.K] = [r.G,r.G] < H. 1+' = r.+.G. 1.- J l.) - l.-) l. J
(iiY Now take H. . = Z. G , J' = 0, 1, ••• , i and K = G. l.-) J 9.
Then for k < j
[H, , ,rkG] = [Z ,G,rkG] < H, '+k = Z, kG. 1-) ) - 1-) )-
(1.18) Theor~: Let R be a ring, P an ideal of R such that pn = 0 for sorne integer n > o. Put K = 1 + P K, = 1 + pi, i = 1,2".. • Then K, is a mUltiplicative 1 '1 group and [K "K,] < K,+, Moreover r,K < K, so that K 1 ) - 1). 1 1 is nilpotent of class less than n. Proo!: Let G be the group of aIl one-to-one
transformations of R onto itself operating on the right. In
particular, G contains aIl the translations T y: x~ X + y, -1 x,y E R, which form an abelian subgroup of G, ~y= ~~y; and also the set of aIl the multiplications by units
b in R vb : x~ xb, x E R. The elements 1 + U, U Epi, i = 1,2, ••• , are units n 2 n-l in R. For u = 0, so that 1 - u + U - ••• ± u = 1 - v is an inverse for 1 + u which is also in 1 + pi ,since n l l v = - U + ••• ± u - is in pi. Moreover, if 1 + u, 1 + u E l 1 + pi, so does the element (l+u) (l+ul )= l+U+ul+uu •
Hence, = {V + ; u Epi} is a subgroup of G, which is K~ l U isomorphic to Ki.
Put Ho=H,and for k > 0 let Hk be the subgroup of H consisting of translations Lq' q E Pk • 'l'hen = Lqu Since for any x E R, v as above
( -1 -1 ) (0 0 ... 0 ) X Lq ° V1+UO Vl +U = x L_q Vl +V "q Vl +U = (x-q) + q(l+u) = 10.
With ~q E Hi-l' u E p this gives [Hi_l,Kr] ~ Hi' so the H, 's and K' satisfy the hypothesis of theorem (1.16). And 1 i i with q E P , U E P we obtain [H"K~] < H,+" all i and j. 1 J - 1 J Moreover K', is the largest subgroup of K' for which this J is true, since ~l E H = HO and for 1 + U E K,
[~l,Vl+u] = ~u E Hj implies u E pj, and this is true only if l+u E Kj' i.e. if
V + E K'j. The results follow then from theorem (1.16). l U Definitions
(1.19) A group G is called solvable if it has a finite series (1.3) such that G + is a normal subgroup of G and i l i the factor groups Gi/Gi +l are abelian, i = 1,2, ••• ,r-l. (1.20) If G has a series like the one above but the factors are cyclic groups, then G is called polycyclic.
(1.21) If in addition to the condition in (1.20) G, is a 1 normal subgroup of G, i = 1,2, ••• ,r, then G is called supersolvable.
(1.22) A group is said to satisfy the maximal condition
(M) if all its subgroups are finitely generated.
(1.23) Corollary of the definitions. For a group G, supersolvable implies polycyelie, and polycyclic implies solvable.
(1.24) Theorem. Let G be a finitely generated nilpotent group. Then Gis supersolvable (henee also polyeyelic). 11.
pr'o'of: Let c be the c1ass of G, so r c+1 G = 1 but rcG ~ 1. By coro11ary (1.13) rcG is a finite1y generated abe1ian group. be a set of generators of r G. c We have
Since rc G is contained in the center of G, a11 the groups in (1.25) are normal in G, and the factor groups are cyc1ic.
App1ying the sarne argument to G/rkG, k = c-1, c-2, ••• , 2,1, we may interpo1ate a series of normal subgroups with cyc1ic factor groups between rkG and rk_1G. Taking a11 these series together we obtain a long one satisfying the requirements of definition (1.21). 12.
§2. POLYCYCLIC GROUPS
(2.1.) Theorem. Suhgroups, factor groups and finite direct products of po1ycyc1ic groups are po1ycyc1ic. Proof: Let G he a po1ycyc1ic group, so it has a series of suhgroups
(2.2.) ..... > G = 1 - r such that G. is normal in G. 1 and G. l/G. is a cyc1ic ~ ~- ~- ~ group, i = 2,3, ••• ,r. (a) Let H he a suhgroup of G, and define H. = H" G.. Then H = Hl > ••• > H = 1 is a ser ies wi th ~ ~ - - r the same properties as (2.2) since
H. 1 H. 1 G.H. 1 G. 1 ~- = ~- ~ ~- < ~- H. Hn G. G. G. ~ ~ ~ ~ is cyc1ic, i = 2, ••• ,r • (h) Now let K he a normal suhgroup of G, and G = G/K and define G. = G.K/K; then ~ ~ G./ G.K G (G + K) G. ~ - ~ i i 1 ~ Gi +1 - = - Gi +1K Gi +1K Gi n Gi +1K which is an epimorphic image of Gi /Gi +1 since Gi +1 ~ Gin Gi +1K, and hence cyc1ic. 13.
Cc>. If Gand H are polycyclic groups with series G G ' > ••• ' > G l, H Hl' > ••• ' > H l, and = l - - r = = - - s = GxH is their direct product, then the series
where t = max(r,s), has the properties of (2.2) and so G x H is polycyclic.
(2.3) Theorem. If G is polycyclic and a torsion group, then it is fini te.
Proof: G has a series (2.2). Since G is a torsion group, the cyclic groups Gi/Gi +l must be finite. Then
r II 1 G. _l/G·I < 00 . 2 ~ ~ ~=
(2.4) Theorem. A group is polycyclic if and only if it is solvable and satisfies the maximal condition (M). (cf. definitions (1.19) and (1.22».
Proof: (a) Assume G is polycyclic. By corollary (1.23) it is solvable and by theorem (2.1) it is enough to prove that polycyclic groups are finitely generated. In fact, if > > 1 is a series such as (2.2) , we can G = Gl ... -G r = choose g. E G. so that G . = < G . + l'g , > for i l, ••• ,r-l. ~ ~ ~ ~ ~ = Then G =
(b) Suppose G is a solvable group satisfying (M). Then there is a series (2.2) with abelian factor groups.
Moreover, Gi (and so Gi/Gi +l ) is finitely generated by hypothesis. Then by an argument similar to that in the proof of theorem (1.24) we obtain a series with cyclic factor groups (though the subgroups in it are not necessarily normal in G). Hence G is polycyclic.
(2.5)· Theorem. Every supersolvable group G has a subgroup H with the following properties: H is nilpotent,
IG : HI < ~, and H > G'.
Proof: G possesses a series satisfying the requirernents in definition (1.19). Let C. be the centralizer ~ of G. l/G. in G/G ; then [G. l'C'] < G .• Since G. 1 ~- ~ i ~- ~ - ~ ~- and G. are normal in G, so is Then G/C. can be ~ ~ regarded as a group of automorphisms of Gi_l/Gi , and hence it is a finite abelian group since G. l/G. is cyclic. Put r ~- ~ H = n C ... Then IG/HI < ~, G/H is abelian (since [g,h] e: C. ~ ~=.. 2 ~ for all i, and all g,h in G) , hence H -> G' and the two last properties hold.
and If we put Hi = Gi n H, then Hr = 1
So H = Hl· > > H is a central series for H, and H is - r nilpotent. 15.
(2.5) Definitions. Let P be a property of groups.
A group G is said to be locally P if every finitely generated subgroup of G has the property P.
G is said to be residually P if for every g ~ 1 in G there exists a normal subgroup K of G such that g t K and G/K has the property P. (2.7) Theorem. (Hirsch). Every polycyclic group is residually finite. Proof: by induction on the number of terms in a series having cyclic factors: It clearly stiffices to prove that if G is a cyclic extension of a polycyclic group H, and H is residually finite,so· is G. 50 let G be generated by H and an element x and let g ~ 1 be any element of G. We have to find a subgroup K of G, normal and of finite index in G, and such that g t K. First assume g t H. If G/H is finite, take K = H. n If G/H is infinite and g - x (mod H), n ~ 0, then we can take K to be the subgroup generated by H and x2n for instance. Next, let g E H. By hypothesis, there is a subgroup
L of H, normal and of finite index in H, and g ~ L. 5ince H/L is fini te, there is a positive integer m su ch that m n = 1 for aIl h in H/Li then Ifl, the subgroup of H generated m by the elements u , U E H, is fully invariant in H and Ifl < L. 16. m In particular g ~~. Now H/H is polycyclic by (2.1), and every elernent of it is of finite order,so it is a finite group (2.3). Moreover x induces an automorphism of H, and also one of H/~ since ~ is fully invariant. Since H/~ is finite, this last automorphism has finite order, say s > 0, so S that X commutes with aIl elements of H (mod ~). S Consider the subgroup N generated by X and ~, it is a normal subgroup of G, and it is of finite index since
s s _~ s [G:!Cl = [G: < H,x >] [
If g ~ N, we take K = N. If g E N, since x S commutes with elements of H mod ~, we have for g an expression g = xks h, h E ~ and k an integer. Since g E H, we have xks E H, so G/H is finite, and also G/~ is finite. We can then take K = ~. (2.8) Corollary. Let G be a finitely generated nilpotent group and let T be the intersection of aIl nontrivial normal subgroups of G. If T ~ 1 then G is a finite p-group. Proof: By theorem (1.24) G is polycyclic, so an element g E T, g ~ l, can be excluded from sorne normal subgroup N of finite index. If n ~ l, then g would belong to N by hypothesisi thus N = 1 and G is finite. G is also nilpotent by hypothesis, so it is the direct product of its Sylow subgroups Si' i = l, ••• ,n, each S. being normal. But then g EnS.,n which is impossible ~ . 1 ~ ~= unless G possesses only one Sylow subgrouPi hence it is a finite p-group. 17.
(2.10) Theorem. Let G be a finite group, and V be the intersection of all normal subgroups of G of prime
00 power index. Then V = ~ , 1 r kG. k=l Proof: (a) Let N be a normal subgroup of prime power index. Then GIN has prime power order, hence it is nilpotent, so there is an integer m > 0 such that G < N. Therefore r m
(b) The subgroup rkG is normal and, since G is finite, GIn rkG is nilpotent and finite; thus
GI f'l r kG is the direct product of its Sylow subgroups
Pl'···'PJI,"
Now Ni = Plooo Pi~lPi+loOO PJI, is normal of index equal to the order of Pi' a prime power and JI, (1 N. = 1. Hence, V < () r kG. i=l ~ 18.
The general references for this section are the same as for the previous one; Theorem (2.7) and the corollary (2.8) can- be found in a paper by Gruenberg [10], and theorem (2.10) was included in that of J. Buckley
[8] • 19.
, '§'3'.' , THE' COLLECTING' PROCE'SS
This section follows closely the treatment of the collecting process of Philip Hall given in his notes [21 and is included mainly for the sake of completeness.
We saw in (1.13) that if G is a finitely generated group with generators al, ••• ,aq , then rnG was also finitely generated mod rn+1G and that a set of generators consisted of all commutators [a , •.• ,a l, where a belongs to the Pl Pn Pi given set of generators of G. Our aim here is to prove that rnG/rn+1G, n = 1,2,... is in fact a free abelian group and to give a basis for it, when G is afree group.
(3.l) Definition. Let A be a set of elements of G, b an element of A. Put A * b equal to the set of elements a a~ [a,b], ••• , a Iak_l,bl, .•• , for all a e: A, a ~ b. = a 0' = k =
(3.2) Lemma. If the group G generated by a set A is nilpotent, then G = .
Proof: Let 9 e: G, so that 9 has an expression in terms of elements of A and their inverses. Then we "collect" b into an element of at the left of such an expression. There are four cases according to the occurrence -1 of b or b to the right of an element of A * b or of an inverse of such element: 20.
(a) g = ••• cb ... , c e: A * b. We replace cb by bc[c,b]; then Ic,b] e: A * band we moved b one place to the left.
-1 (b) g = ••• c b ••• , c e: A * b. We use the
When moving b -1 we use the identities
-1 (c) cb = b -1 c[c,b-1 ] and
(d) c -1 b -1 = b -1 [c,b -1 ] -1 c -1 ,
so we have to prove that if c e: A * b, then lc,b-ll e: . -1 Put Co = c, ck = [ck_l,b], c~ = [ck,b l. Since -1 * b-1 1 = [ck,bb 1 = ck ck+1 by (i) of Lemma (l.l),we have
from which (c*)-l = and k Then
(c*)-lc-1 = 1 1 ......
Since ck' c~ e: rk+1G and G is nilpotent, this process gives a finite expression of [c,b-1 l in terms of e1ements of A * b
and their inverses. This completes the proof.
Let X be the set X = {x , ••• ,x }, q > 1, and let 1 q [u,v] be a formaI operation, the "commutator" of u and v. 21. (3.3 ) Define C = CCX} to be the smallest set of commutators which contains xi' i = 1,2, ••• ,q and with u and v also contains [u,v]. We define the weight of an element in C by the rules wt x. = l, i = 1,2, ••• ,q, and wt[u,v] = wt u + wt v. ~ (3.4) A basic sequence is a sequence b ,b ,b , ••• , l 2 3 of elements in C satisfying : (a) b. = x., i = 1,2, ••• q ~ ~ (b) for k > q define bk to be an element of minimal weight in Xk- l , where Xo = X and for k > 0 Xk = Xk- l * bk • From (a) and (b) it follows that i < j implies wt b .. ~ wt b. ~ J Thus bl, ••• ,bq are all the commutators of weight 1 and if Xk has an element u of weight equal to wt bk , put bk+l = u. Otherwise, take bk+l to be any of the elements in Xk of weight equal to wt bk + 1. (3.5) Remark. By the construction we see that for any n there is a k such that Xk contains no element of weight less than n. In the discussion that follows, we will work with a fix basic sequence. 22.
(3.6) The rnembers of this fix basic sequence will be called basic commutators.
(3.7) Definition: For i > 0 let P. be the set ~ of formal products r l r 2 ri Pi = bl b2 bi qi' rj E z. wllere r j ~ 0, j = 1,2, ••• ,i and qi is a product of elements of X .• ~ The weight wt p. of such an element is the surn of ~ the weights of the factors. (3.8). Lemma. There is a one-to-one correspondence
~i between Pi and Pi +l , i ~ 0, which is weight preserving. Proof: Let P. be as in (3.7), so that q. is of ~ ~ the form t 2 sn (3.9) q. c b. 1 c b. 1 n ~ 0, ~ . . . . . • 2 ~+ ••• n ~+ ,
the c j also belong to the Xi +l , and the ••• represent a product of elements of Xi different from bi +l • Define ~i (Pi) to be that element Pi+l of Pi +l t. obtained by replacing in Pi the products Cjbi~l' j = l, ••• ,n by the commutator c. [tj ] = [ ••• l [cj ,bi +l ] , bi +l ] ••• ] wi th t j J terms bi +l , which is in Xi +l if c j E Xi' c j # bi +l • 23.
Conversely, if [t ] l and = ••• cl .. · > 0, n ~ 0 and c j € Xi' c j ~ bi +l and the ••• stand for products of elernents of X. different frorn l.
which is of the forrn (3.7) with
Then ~. and cp. are inverses, hence is one- l. l. to-one and onto Pi +l , and from the definitions it is clearly weight preserving.
(3.10) Definition. Any finite product
for sorne N, r. > 0 , l. - is called a basic product in the basic conunutators bl ,b2 , •••
(3.11) Rernark. By the rernark (3.5), for any w there is an integer N = N(w) such that XN contains no elernent of weight ~ w. Thus the elernents of Pi" of weight w are all basic products if i ~ N(w).
(3.12)" C"or"o"l"l"ary. The nurnber of basic products of w given weight w is q •
Pr"oof. The elernents of weight w in Po are the products Since CPi' as defined in the proof of 24. lemma @.8), is one-to-one and weight preserving, the images of thèse elements under ~i-l ••• ~o must be exactly the elements of weight w in Pi' and these are aIl basic products if i ~ N(w). Moreover any basic product of weight w must belong to Pj with j < N(w) since bk has wt bk" > w for aIl k ~ N (w) •
(3.13) Thefre"eassociative ring in xl' ••• ,Xq.
Let ~ be the set of aIl linear combinations with coefficients in Z, the ring of integers, of the qW products x ••• X and let RO = Z. Put R = l Rw, direct sumo Pl Pw w=O With the obvious multiplication (juxtaposition of the generators, extended by linearity) R becomes a ring, the free associative
For u,v in R define [u,v] = uv - VU1 we then can consider the set C defined in (3.3) and a basic sequence Then a basic commutator b. of weight s belongs 1. to Rs' and the basic products of weight w belong to ~. Moreover we have the following
(3.14) Lemma. The qW basic products of weight w forro a basis for the free Z-module ~.
Proof: Since the rank of Rw is also qW, it is enough to prove that the basic products forro a set of generators. We show that every element Pi E Pi can be 25. expressed as a linear combination of elements of Pi +l of the same weight. Then the elements x x of the basis Pl· •• Pw of Rw' which are the elements of Po of weight w, will be linear combinations of elements of the same weight in Pi' ~ll i, ~nd these elements are the basic products of weight w if i ~ N(w), as we remarked before. For any u,v in R we have, taking the commutator of u with v,r times, r [u,v, ••• ,v] = uvr - rvuvr-l + (2) v 2 uv r-2 . . . or .. uvr = [u,v, ••• ,v] + rvuvr - l - (2) v 2 uvr-2 + •••
If now P. is as in (3.7) with q. as in (3.9), . 1 1 S we use the formula above to replace each Cjbi+l by a linear combination of products of the same weight in each of which the number of bi~l's to the right of elements c in Xi is smaller than before. Note that since c j € Xi' c j ~ bi +l we have that [cj,bi+l, ••• ,bi+l] € Xi +l • After a finite number of steps we will have qi expressed as a linear combination ri+l of elements of the form bi +l cl ••• cp' c j € Xi +l , and Pi as a linear combination of elements of Pi +l of the same weight as p .• 1 (3.15) Corollary. The basic commutators bl ,b2 , ••• are linearly independent. 26.
Next, take an integer n > 1 and consider the . S R/pn, h r1ng = = w ere E is the ideal Rl + R2 + ••• in R. The set of elernents l+h, h E p/pn in S for.m a nilpotent = = group of class < n, c.f. theorern (1.18); let H be the subgroup generated by the elements y. = l+x., i = 1,2, ••• ,q. 1 1 Now we use the operation [u,v] = uv - vu in S to forrn a basic sequence b ,b , •.. in {x.; i = 1,2, ••• ,q}, and the l 2 1 operation [u,v] = u-lv-luv in H to obtain the corresponding basic commutators c ,c ' ••• in the {y. = l+x.; i l, ••• ,q}, l 2 1 1 = so that for example b = [x.,x.] if and only if k 1 J
Let b be the last basic commutator of ~eight N less than n. Then bN+l ,bN+2 , ••• belong to Rn + Rn+l + ••• = fn and so they represent zero in S.
Similarly cN+l ,cN+2 ' ••• are commutato~s of weight n in H, = We have· the following relation so they are in r n H 1. between the two sequences:
b. i (3.l6) Lemma. Let wt c.1 = wt 1 = w, l < < N; w+l then Ci - l+b rnod p . i =
Proof'.by induction on w. If w = 1 we have by definition c. y. l+x. 1 = 1 = 1 = l+bi •
,b ] , c~ with For w > 1 let b i = Ib j k 1 = Icj,ck] wt b wt c. = r, wt b = wt c s and r+s = w. By j = J k k = induction hypothesis we have c. - l+b. mod pr+l, -' J J = 27.
Now Cj,Ck e: H = so they are of the form c = 1 + a , c = 1 + a b. mod pr+1, j j k k J <= - b d s+l Th a k = k mo ~ • en
r+s+1 where . . . denote terms in g , and
so that
mod r+s+1 ~
(3.17) Lemma. Anye1ement h e: H =
Proof: We app1y 1emma (3.2) with b = c1 ,c2 , ••• ,cN to obtain H =
Suppose
••• ,V _ but ~ 1.1 .; let wt c. w and let i 1 = 1.1.1.- 1 vi 1. 1. = c,+l ,···,c'+k be the remaining terms of weight w. Then 1. vi 1. v N l.1i l.1 N v = ci ••• c c. ... c and by the preceding 1emma N = ~ N 28. and also k w+l v - 1 + L ~.+ mod p ~ a a=O
But by corollary (3.15) this is impossible if vi ~ ~ii thus the lemma follows. (3.18) Theorem. Let F be the free group on the generators Yl, ••• ,Yq • Then H = F/rnF. Proof: Let ci'- y-'- , y n be the elements and subsets of F obtained in the same way as ci'Y'Yn ' but using the Yi' Consider the homomorphism e : F ~ H determined uniquely by mapping Yi ~ Yi' i = 1,2, ••• , qi it is onto H. Let K be its kernel, so that F/K = H. Since H is nilpotent of class less than n, so is F/K, thus r F < K. As above/applying lemma n (3.2) repeatedly to the nilpotent group F/rnF, we obtain r F,or - n F =
èontrary to lemma (3.17) (3.19) Corollary. Let F be the group freely generated by Yl'." Yq , and let ël ,ë2 , ••• be a basic sequence in these elements. Then for any integer n > 0 the quotient rnF/rn+1F is a free abelian group with a basis given by the - cosets of those c j with wt ë. = n. J 29.
'pr'oof: It fo11ows from (3.18) since in that theorem n was chosen arbitrari1y.
(3.20)' Theorem. Let F be free on q generators.
The rank of rwF/rw+1F is
n(w,q) ~; l ~ (d)qw/d dTw where ~ is the Mëbius function, defined by:
if d is the product of r > 0 different primes, p (d) = l o otherwise.
Proof: We know that the nurnber of basic products w of weight w is q • A1so, the basic products of weight w forro a basis for,the free abe1ian group Rw. These are the ~1 ~n e1ements = b b with i wt b = w. Put b~ 1 n l ~ i w wt b i = di,so q is the number of sequences ~ = (lJ l' ••• ,~ n) such that .d. = w. If we let A(t) = l qWtW, this is l ~ J. J. equiva1ent to
co co 2d d. i 1 1 (1 + t J. + + ) 1-qt = A(t) = TI d. = TI t . . . i=l i=l (l-t J.) since the coefficient of t W is exact1y qW in A(t), and the number of such sequences ~ in the series in the right hand side. 30.
Now, for any integer w, we have that in the 00 d. 1 product rr (l-t)J. - the number of factors with d. = w i=l J. is the rank n(w,q), since both are equa1 to the number of basic commuta tors b. such that wt b. = w. Hence J. J.
1 00 1-qt = rr 1 w=l .' (l_tw)n(w,q)
Taking 10garithms we obtain 00 l log 1 = log 1 1-qt w=l (l_tw)n(w,q) . Since 00 log 1 = l 1 t n , 1-t n=l n we have 00 00 00 Wv 1 qntn n(w,q) l 1 t • l -n = l v n=l w=l v=l
Equating coefficients we have for each n
n -1 q = l n(w,q) -1 n wv=n v or n q = l n n(w,q) = l w n (w,q ) . wv=n v wIn 31.
We use now Mëbius inversion formula: if 'g(n) = l f(d) dTn then f(n) = III (n/d)g(d). Thus we obtain, with 9 (w) = qJ w, dTn f(w) = wn(w,q)
wn(w,q) = l II (w/d)qd = l II (d)qw/d. dTw dTw 32.
§4. GROUP RING AND DIMENSION SUBGROUPS
The results given here are well known, and in the literature: they are concerned with properties of the group ring and dimension subgroups of a group Gi we try to give simple but complete proofs of them. We begin with the- appropriate Definitions. Let G be any group, Z the ring of integers.
(4.l~) The integral group ring ZG is the free Z-module on the set of generators{gi 9 E G} with the multiplication induced by linearity from the multiplication in G. (4.2.) The difference ideal of the group ring of G is the two sided ideal 6.G generated by {g-li 9 E G}. Equivalently, one can define 6.G to be the kernel of the augmentation homomorphisme
(4.3.) E : ZG + Z defined by
E ( Zgg) = Z E Z , ï ï 9 gEG gEG where Z E Z and are all zero except for a finite number. 9 For this reason 6.G is also called the augmentation ideal.
The equivalence is easy to check: if 33.
a;::; ~ cr (g-l) e: âG, a e: ZG, g~l 9 9 then e: (a) = L e:(crg)e:(g-l) = 0 sinee e: (g) = 1 for any 9 e: G. So âG < Ker e:.
On the other hand, if e:(a) = 0, where a = L Z 9 is ge:G g in ZG, then ~ Z = 0 and ge:G g
cr = L Z 9 - ( L Z )1 = L Z (g-l) ge:G 9 ge:G g ge:G g
belongs to âG. So Ker e: ~ âG, and we proved the equivalenee.
(4.4) Remark. Given any group homomorphism ~ of G
onto H, it induees a ring homomorphism ~* of ZG onto ZH
whieh maps âG onto âH, beeause ~*(g-l) = ~(g) - ~(l) = ~(g) - 1.
Also for any r ~ 0, (âG)r is mapped by ~* onto (âH)r
sinee ~* is a ring homomorphisme
(4.5) Lemma. For any group G, âG is a free abelian
group on '{g-l~ g e: G, 9 ~ l}.
Proof: If we had L Z (g-l) = 0, Z e: Z, then g~l 9 g (~ Zg>l= O~ sinee ZG is free on{g~ g e: G}, this
implies that Zg = 0 for aIl g ~ 1.
We study a bit more elosely the integral group ring and differenee ideal of free groups.
So let F be the free group on the free generators
'{Xi~ i e:I}. First note the following identities in ZF: 34.
(4.6) (fl-l) + (f2-l) + (fl-l) (f2-l) = (fl f 2-l), f l ,f2 E F;
(4.7) f-l_ 1 = -(f-l) + (f-l)2 + ••• + (_l)n(f_l)nf-l, n > 1. any f E F;
Proof of (4.7). For n = l -(f-l)f- • Assume n > 1 and (4.7) true for all k < n. Then
= -(f-l) + ...
Proof of (4.8). We compute the right-hand side:
x ([fl ,f2] - 1) =
-1 -1 = f l f 2- f 2f l - (f2f l -l) (fl f 2 f l f 2- 1) =
-1 -1 = f l f 2- f 2f l - f l f 2 + f 2f l + f l f 2 f l f 2 - 1 =
Also every cr E ZF can he expressed uniquely in the forro
(4.9) cr = ZQ + l Z, (x.-l) +. l Zij (xi-l) (xj-l) + ô iEI ~ 1 1,jEI 35. with and here are zero except for a finite number.
In fact, we have the more general
(4 .10)' The'o'r'em. Let F be the free group on generators{xi ; i € I}. Then
(6.F) i ZF == Z, and i = 1,2, ••• 6.F (6.F)i+1 ' is afreê Z-module.
Proof: (a) Since every element f € F is a finite product of generators and their invers~s, formulae (4.6) and (4.7)show that the difference ideal 6. = 6.F is generated by
{(xi-l)~ i € I} as a left (or right) ZF-module. We now prove that in fact it is a free ZF-module on the sarne set of generators.
To prove this is to prove that any map
where W is sorne left ZF-module, can be lifted to a
ZF-homomorphism fi : 6.F + W.
First fOrIn the 'sp'lit 'extension U of W by F, i.e. the group of pairs (f,w) € F x W with the following multiplication rule:
(f,w) (g,u) = (fg,w+fu). 36.
We check that it is associative:
(f ,w) «g, u) (h, v» = (f ,w) (gh, u+gv) = (f (gh) ,w + f (u+gv»
( (f ,w) (g, u) ) (h,v) = (fg ,w+fu) (h',v) = «fg) h,w+fu+ (fg) v)
-1 -1 and that the inverse of (f,w) is (f ,-f w):
since clearly (1,0) is the identity in U.
Next, let k: {xi; i E I} ~ U(F,W) be the map given by
Since F is free on '{x. i i El}, this map lifts to a 1.
homomorphism k : F ~ U, which is the identity on U/W (i.e. on
the first coordinate). Define a map h : {(f-l); f E F} ~ W by the formula
k (f) = (f,h (f-l) ) •
Now 'if-li f E F} is a Z-basis for the free Z-module ~F (cf. lemma (4.5», so that h can be extended to a
Z-homomorphism from ~F into W, which we also denote by h.
We have k(fg) = k(f)k(g) since k is a group homomorphism; then
k(fg) = (fg,h(fg-l» = (fg,h(f(g-l) + (f-l»
= (fg,h(f(g-l» + h(f-l» 37. and
k (f) k (g) =:= (f,h(f-l» (g,h (g-l» = (fg,h(f-l) + fh(g-l»)) give h(f(g-l» = fh(g-l) so that h is actually a ZF-homomorphism; hence ÂF is a free ZF-module. (b) ÂF is an ideal in ZF with the property that ZF/ÂF - Z, cf. (4.3). Let  = ÂF and denote ÂF.M by ÂM. Now, if M is any free left ZF-module, then Â: is a free (Z~) - modulë, hence a free Z-module by the previous remark. In fact, consider the homomorphism from
ZE' ~ M cnte ~ZF
induced by the projections ZF ~ ZF/ and the identity on M. It is easy té> check that the kernel is  @ M; moreover, this module can be identified with the submodule ÂM of M. Hence M/ÂM ~ (ZF/Â) ~ZFM, and the la st module is a free (ZF/Â)-module since M is free over ZF. Now take M = Â; then Â/ 2 is a free Z-module on {x.-l; i El}. l. We now consider Â2; as an ideal of ZF it is generated by {(x.-l) (x.-l); i,j El}. We prove that Â2 is J. J a free ZF-module in the same set of generators. 38.
For if cr.. e: ZF, i, j e: l and 1J
we must have L cr. . (x. -1) = 0 for all j e: l since 11 is i 1J 1 free in for the same reason cr .. = 0, all i 1J and j.
2 Again, put M = 112; then 11 is a free Z-module
The same type of argument shows that in general 11~ is a free Z-module (mod 11~+1), and a set of free generators is
{(xi - 1) ••. (x. -l)~ il, ••• ,i~ e: Il. 1 1~ Now we are in a position to introduce the principal characters of this story :*1 the dimension subgroups.
(4.11) Definition. Let G be any group. Then
is called the n-th dimension subgroup of G.
and D n G _.< D n-lG.
Moreover DnG is normal in G, because it is the kernel of the following group homomorphism
G ~ ZG ~ ZG/(I1G)n
*) Cf. 14J, page 135. 39. which maps 9 into 9 + ~n, where the first arrow is the inclusion and the second one the natural homomorphism of ZG cnte the quotient.
(4.l2) Lemma. Let N be a normal subgroup of G. Then the sequence
'If o + I{N) + ZG + Z{G/N) + 0 is exact, where I{N) is the left ideal generated by
{g-l; 9 EN}, and 'If is induced by the canonical homomorphism
'If : G + G/N.
Moreover, 9 E G, (g-l) E I{N) imply 9 E N.
Proof: Let cr = 2 Z g, Z E Z, be in ker 'If. Then if we 9 9 9 denote 'If Cg) = g,- we have
'If (cr) = 2 Zgg- = 0 = 2 { 2__ Zg)h , fiEG/H g;g=h which implies (2 z) = 0 since Z{G/N) is a free Z-module g=fi 9 on{g; g E GIN}. Let fi be a fixed element of GIN, and
~ , denote by 1.. the SUIn·. over . {g; g = h}. Consider the element , 2 Zgg, which by what we just proved is equal to
-1 , and since .fi = g, we have h 9 E N; thus 2 Zgg belongs to the ideal I{N), and so does cr, which is a SUIn of such elements. 40.
On the other hand, it is clear that the elements g-l, 9 E N, all be10ng to ker~, so ker ~ = I(N) and the sequence is exact.
Moreover if 9 - 1 E I(N) and 9 E G, then 9 E N since ~(g-l) = 0 if and only if 9 = ï = N.
(4.13) Theorem. For any' group G, and any pair of
~o~ittye integers n,ID we have
In particu1ar the series
(4.14) > DG> - n ... is a central series for G.
'Pxo'of: It is enough to prove that
since !g,h] is agenerator for [DnG, DmG1.
By (4.8), which is true even if F is not a free group, [g ,h] - 1 :: (g-l) (h-1) - (h-1) (g-l) mod (~G) n+m
But 9 - 1 E (~G)n and h - 1 E (~G)m so that [g,h] - 1 - 0 mod (~G)n+m.
In particu1ar for m = 1 we obtain 1DnG,Gl ~ Dn+1G and (4.14) is a central series. 41.
n = 1,2,3, ••••
Pr'oof: It is c1ear since r n G is the lower central series (cf. (1.7». 42. §5. THE POINCARE-BIRKHOFF-WITT THEOREM
Here we introduce the notion of a Lie algebra ~ and its universal algebra lA, and obtain the result that J! is ismorphic to a subalgebra of lA L. For a thorough exposition on the subject of Lie algebras one should refer, e.g. to N. Jacobson [4]. Throughout this and the next sections, let R be a commutative ring with identity. All algebras and modules will have structure ring R. We freely inter- change the terrninology of vector spaces and of free R-modules. (5.1.) A free R-module J! which is an algebra over R is said to be a Lie algebra if the multipLLcation in J! , denoted by [x,y], x,y E J! , satisfies
(a) [x,x] = 0 for all x E J! i (b) [[x,y],z] + [[z,x],y] + [[y,z],x] =0 for all x, y, Z E J!, called the Jacobi 1 s identi ty.
In the case that char R ~ 2, (a) is equivalent to
(al) [x,y] = -[y,x], x,y EJ!. (5.2.) Exarnple. Let A be an associative algebra. We denote AL the algebra obtained from A by defining a new multiplication [x,y] = xy - yx, called the Lie product
, 43. or the commutator of x and y",x,y e: A. With the new product AL is a Lie algebra.
(5 .3)' De'fini'tion. A universaI ,( enveloping or ass'ociat'ive) 'al'gebra of a Lie algebra S, is a pair (1.4 , 1) , where 1.4 is an associative algebra wi th an identi ty and
1: S, + Ù L is a Lie algebra homomorphism, satisfying the following universal property.
(VI) If A is any associative algebra with an identity and 9 : S, + AL is any Lie algebra homomorphism, then there is a unique homomorphism 9' : 1.4', + A such that
9 = 19'.
In a diagram / / 9' / ~
If it exists, 1.4 is unique up to unique isomorphism for if (1.4',1') is another universal algebra for- S, , we put A = 1.4', 9 = l'in (Ul), and then reverse the roles of (1.4,1) and (~,l'). Because of this uniqueness, one often refers to "the" universal algebra of s,.
We now prove the existence of the universal algebra for any Lie algebra s,. We recall the following
(5.4) Defi'n'i'ti'on. Given any R-module M, we define the tensor algebraon M by 44.
where M -M 1'"" and Mi = M ® M ® ... ® M i times. MUltiplication in T(M) is given by
(5.5) (ml@ ••• ®mi)®(nl® ••• ®nj ) =
= ml ® ... Qg mi ® nI ® .•. ® n j
for ~, n· h e: M, k = l, ••• , i , h = l, ••• , j .
The tensor algebra has the property that any linear mapping ~: M ~ A, where A is any associative algebra, has a
unique extension ~ : T(M) ~ A.
Another property of the tensor algebra is that it is a graded algebra in the sense of the
(5.6) Definition. An algebra A is said to be graded if it is the direct suro of subspaces A. , i = 1,2, ••• ~ such that
00 A = A" A,A, < A,+" aIl i and j , l 0 ~ ~ J - ~ J ~=
Now let T = T(l) be the tensor algebra of the
underlying vector space of l, lët ~ be the two sided ideal of T generated by the elements of the forro
[x,y] - x ® y + y ® x, x,y e: l, and let lA = T/~. Finally let
t be the restriction to : of the canonical hamamorphism of T onto lA. We have 45.
(5.7)' The'o'rem. ( tu t) is a universal algebra for t.
'Pr'oof: 'Let A be any associative algebra. Gi ven
S : t ~ AL there is a unique extension S : ~ ~ A, mapping xl ® ... ® xi into S (xl) ••• S (xi). Hence
.... S([x,y] - x ® y + y ®x) ;::: S([x,y]) - 9(x)9(y) - 9(y)9(x) = 0, since 9 is a Lie algebra homomorphism, and so 9 also maps
~ into {a}. Then we can define a homomorphism 9' : ~/~ ~ A such that S' t = 9. The uniqueness of follows from that of .... 9 ' e . Next we give a description of a basis of lA starting with an ordered basis '{Xii i e: I} of t , l an ordered set of indices. The monomials
(5 • 8) x p ®... ® x P l n
, ' i~br{ a basis for tn = t ® ••• ® t, n times, n > 1.
A monomial such as (5.8) for which
Pl < P2 < ••• ~ Pn is called a basic monomial.
We define the ind'ex of the monomial (5.8) by ind (x ®. .. ® x ) = l: y. k' where ~y ik = 0 if Pl Pn i Yik ;::: l if Thus, a monomial is basic if and only if its index is O. Moreover we have, if Pk > Pk+l 46. ind (x ®. .. ® x ) = 1 + ind (x ®... ® x ® x ® ... (j) x ). Pl Pn P.l Pk+l Pk p n One ,can define a partial order in the set of all monomials using the degree (i.e. length), and in the set of monomialsof a given degree, by using the index. We state now the (5.9) Theorem. (Poincaré-Birkhoff-Witt) The cosets of 1 and the basic monomials forro a basis for ù = T/~. The proof is divided into two lemmas. (5.10) Lemma. Every element of T is congruent mod ~ to a linear combination of 1 and basic monomials. Proof: It is enough to prove the lemma for monomials such as (5.8), and we use induction with respect to the order given above. So let us assume that the lemma holds for all monomials of degree < n and all monomials of degree n with index < ind (x ® ... ® x ). Suppose Pk:> Pk+l. Then Pl Pn x ® ... ® x = x ® Pl Pn p 1 - x ®. .. x ® x . •• ® x + x ® ... ® [x ,x J® ••• Pl Pk+l Pk Pn p 1 p k' p k+l ... ® x p mod ~ n since x ® x- x ® x - [x ,x J e: ~ by definition. Pk Pk+l Pk Pk+l p k p k+l 47. Now the first summand in the last expression has lower index, and the second one is a linear combination of monomials of lower degree. So the lemma follows by induction. (5.11) Lernma. Let Vn be the vector space with basis '{x x ••• x ; Pi E I, Pl ~ P2 ~ ••• < Pn} and let Pl P2 Pn V = R.l$Vl E9V2ct> ••. Then there exists a linear rnapping ~ T ~ V such that (a) ~(l) = 1; (b) ~ (x Qg ••• ® x ) = x ••• x if Pl < ••• < P , and Pl Pn Pl p n - - n (c) ~ (xp ® ... ® x - x ® ... x ® x p ® ... ® x ) 1 p n p 1 p k+l k p n = ~ (x ® ... ® [x ,x ] ® ... @ x ). Pl Pk Pk+l p n Proof: Let l . be the subspace of ln spanned n,~ by the monomials of index < i. Define ~(l) = l, and suppose ~ has been defined on the subspace RI (f) ••• Define ~ on l 0 by ~ (x ® •.. @ x ) = x x x , n, p 1 p n Pl P2 Pn since x @ •• €)x has index 0, i.e~ is basic, and Pl Pn extend by linearity. Next suppose ~ defined on Rl e ... œln, i-l' i > 1. We want to define ~ on a monomial of index i. Let (5.8) be such a monomial; assume Pk > Pk+l' and put (5.12 ) ~ (x ® ... ® x ) = ~ (x ® ... ® x ~ x ® ... ® x p ) Pl Pn p.1 p k+l p k n + ~ (x ® ••• ® Ix ,x ] ® ... ® x ); Pl Pk Pk+l p n 48. the two terms in the right-hand side are defined by induction hypothesis. We still have to prove that (5.12) is we11 defined, i.e., the right-hand side does not depend on the integer k, 1 < k < n such that Pk > Pk+1. Suppose there is another integer j, 1 ~ j < n with Case{i) j > k + 1. Put x = u, x = v, Pk Pk+1 x p. = x, x = y. Then using (c) severa1 times we get J Pj+1 $( ••• ®v®u ... @x@y ... ) + $ ( ••• ® [u, v] ® ... ® x ® y ® ... ) = $( ••• v@u ® y @ x ••• ) + $( ••• v ® u ••• OS> [x,y] GSl ••• ) + $( ••• ® [u,v] ® ... ®y ®x ••• ) + $( ••• ® [u,v] ® ..• ® [x,y] 6j) . . . = $ ( ••• u ® v ••. ® y ® x ••• ) + $ ( ••• u ® v @ ••• ® [x,y] ® ... ) . Cas'e (ii) k + 1 = j. Again put x = u, x = x p. = v, x = w. Starting wi th the Pk Pk+1 J Pj+1 defini tion-, (5.12) for the pair k, k+1, we have $( ••• @uQ!)v@w ••• ) = W( ••• v®u®w ••• ) + W( ••• ® [u,v] ®w ••• ) = W( ••• v®w®u ••• ) + W( ••• v® [u,w] ® ... ) + W{ ••• [u,v] @w ••• ) = W( ••• w (il v ® u ••• ) + W( ••• Ci> [v,w] ® u cID ••• ) + + W{ ••• v ® [u,w] ® ... ) + W( ••• ® [u,v] ® w ® ••. ) and from (5.12) for the pair j, j+1 we obtain 49. W( ••• &)u®v®w@ ••• ) = W( ••• u®w®v ••• ) + + W( ••• u ® [v,w] ® ... ) = W( ••• w ® u ® v ••• ) + W( ••• ® [u,w] ® v ••• ) . + W( • •• u ® [v, w] ® ... ) = W( ••• ®w @v ®u ••• ) + W( ••• @w® [u,v] ® ... ) + W( ••• ® [u,w] (3) v ® ... ) + W( ••• u ® [v,wJ ® ... ) . To prove that these two expressions are the same, is to prove that W( ••• [v,w] ®u · .. ) - W( •• • u@ [v,w] ® ••• ) + W( ••• v ® [u,w] · .. ) - W( ••• [u,wJ ® v ... ) + W(. • • [u,vl®w · .. ) - w( .•• w ® [u,vJ ... ) is zero. But by induction hypothesis (c) holds in R (f) ••• <.T) 1. n,1-0 l' hence the ab ove sum is equa1 to ( w( .. •• ® [[v, w] , u] ® ... ) + W( • •• ® [v, [u, w]] ® ••. ) + w( ••• ®[[u,vj,w] ® ... ) = w( ••• ® (lIv,w],u] + I[w,u],v] + [[u,vl,w]) ® ... ) and this is equa1 to zero by the Jacobi's identity. Hence we may extend W to 1. n,10' and the 1emma is proved. To complete the proof of the theorern (5.9) we observe that Lernrna (5.10) says that the co sets of 1 and the basic monomials span Ù = L/~. By (c) of Lemma (5.11) we have that 50. the ±mage of ~ by ~ is '{O}, so that ~ factors throug~ T/~. Now, the images of the basic monomials form a basis for V, so the:.basic monomials themselves must be linearly independent. This finishes the proof. (5. 13) Cor·ol"lary. The mapping \ l .... ù is injective, and R.l n \(l) = O. Proof: We proved that for a basis '{xi ; i e: I} of ) = l, the cosets of 1 and \(xj x j are linearly independent. Both statements are then clear. 51. § 6 • MAGNUS THEOREM The main theorem of this section was first proved by Magnus in 1937, and can also be found in the book by Magnus, Karrass and Solitar [3]. The proof we give here is a modified version of the one presented by J.-P. Serre [5]1 Serre's definition of dimension subgroups is based on the algebra of formaI series, but as we see at the end of our proof, it is equivalent to the one we have given, based on the group ring, which is more convenient wh en dealing with non-free groups, (cf. [2], [7], [8], [11] ). The method described in this section can be used not only for free groups; for instance J.P. Labute [9] applied a generalization of it to the study of the Lie algebra associated with the lower central series of a group with one defining relation; he showed in particular that for such groups the lower central series and the series of dimension subgroups coincide. The notion of a free algebra on a set X of generators can be formulated in a similar way to that of the universal algebra of a Lie algebra. Definitions. Associative algebras are assumed to have an identity. (6.1) A pair (~,i) is called a free associative algebra on the set X of generators if ~ is an associative 52. algebra, i is a mapping from X to ~, and for any mapping a : X ~ A of X into an associative algebra A there exists a unique homomorphisrn a' : ~ ~ A such that ia' = a. (6 .2) A 'f'r'e'e Lie 'al'gebra on X is defined to be a pair (;;l,i) where ~oC is a Lie algebra, i : X ~ ~ oC and given any rnapping a : X ~ B, B a Lie algebra, there is a unique homomorphism a' : ~oC ~ B such that ia' = a. In either definition, if such a free algebra exists, it is unique up to unique isomorphisrn, and we will see from the construction that i can be regarded as an inclusion, and a' as an extension of a in both cases. The existence of ~ and ~oC can be proved similarly to that of the universal algebra. Let M be the free R-module with basis X, R an associative ring with identitYi then it is enough to take ~ = T(M), the tensor algebra of M, and i to be the compasition of the inclusions of X into M and of M into T(M). Moreover, given a ': X ~ A, A an associative algebra over R, one can extend a to M, since M is free on X, and then to ~. The free associative algebra on a set X is thus also a graded algebra, with hornogeneous n-th component ~n = M ® •.. ® M n times. 53. This construction provides explicitly a basis for 3 in terms of the elements of X. It is more difficult to obtain such a basis for the free Lie algebra; but the existence of:J: follows easily from' the above. Let 3: be the Lie subalgebra of 3 L generated by the subset X. Let B be any Lie algebra and (~,l) its universal algebra. If 8 : X + B is any mapping, then 18 is a mapping of X into ~; hence, 18 has an extension 8 : to 3, the free associative algebra on X which can be considered as a Lie-homomorphism 8 : 3 +~ Since 8 L L. maps X into B, the image of 3: lies in B, so we have constructed an extension of 8 to 3J:. Uniqueness follows from the fact that X generates~by definition. (6.3.) Theorem. 3 is the universal algebra of 3J: • Proof: We use the same type of argument. If 8 ::JJ: + BL is any mapping, B an associative algebra, consider the restriction of 8 to X. As before, it has an extension to 3 which must coincide with 8 on 3J: , since 3J: is generated by X. We will also need the notion of the algebra 3 arising from 3 by allowing infinite sums. (6.4. ) Definition. An element ~ e: 3 is represented by a formaI series 54. co S ;:,. .. t sn' where ;n e: 3 n • n=O ~ is an associative algebra with the usual product of series: co if n = 2 nn is another element, we have n=O co ;n = 2 ( 2 ~pnq). n=O p+q=n One can consider -~ as the completion of ~ in the topology given by the filtration . {~(n) = l..t ~ .,• n = 0,1, ••• }, i>n l. but we will not need it. We study now the way one can associate a Lie algebra with a filtration on a group. (6.5) Definition. An integral filtration on a group G is a map v : G -+ Z U' {+ co}, Z the ring of integers , satisfying (a) v (1) = +co ., (b) v(g) > a for aIl 9 e: G; (c) v(gf-l ) ~~iri{v(g), v{fl}, aIl f,g e: G; (d) v([g,f]) ~ v(g) + v(f) for aIl f,g e: G. 55. (6.6)' E:xample. If Gn is a descending central series of subgroups of G, 50 that (i) G = G, l < (ii) Gn+1 Gn , and we define v(g) = supin; g € Gn}. It is a filtration, for 1 € Gn , aIl n, so v (1) = +00 and for aIl g € G, g € Gl so that v(g) > 0; -1 -1 also g € Gn if and only if g € Gn , hence v(g) = v(g ). Now if v(g) = n, v(f) = m and say, n ~ m, then Gn :: Gm -1 v(gf-l ) so that gf € Gn and > n = inf{v(g) ,v(f)}; moreover [g, f] € Gn+m by (iii) so that v([g,f]) > n + m = v(g) + v(f). This is a typical example since we have the following (6.7) Theorem. AlI integral filtrations of the group G come from a descending central series, and the correspondence is one-tc-one. Proof: Given a filtration v as in definition (6.5), we .t:>ut for n' > 1 G {g € G; v(g) > n} • n = 56. Conditions (a) and (c) show that Gn is a subgroup and (b) irnplies Gl = G, whilJ,;e" (d) implies (iii). Clearly the filtration defined by '{Gn ; n > l} coincides with v, and vice versa. (6.8) Defini'tion. Given a filtration on a group ,G, or equivalently a descending central series Gn , n = 1,2, ••• , we define grG is called 'the graded Lie algebra'associated withthefiltrati'on v, or with the central series '{Gn~ n ~ l} (see theorem (6.9) below). grnG is an abelian group, because [Gn,Gn] ~ ,Gn+n ~ Gn+l for all n. Also since IGn,G] ~ Gn+l , one has (~) = g, all f,g E G, where 9- represents the class of 9 in grnG, n = v(g). ( 6 • 9 ) Theorem. The maps y n, m : Gn x Gm -+ Gn+rn defined by (g,h) -+ Ig,h] induce a structure of graded Lie algebra in grG, over the ring of integers Z. Proof: 5ince grnG is abelian, it has a natural structure of Z-module with "addition" given by the product, i.e. 9 + f = gf. 50 if x,y E grG, gi,hi E Gi , gi = xi' hi = Yi' and 57. (6.10 ) y. , x == ? y = l 1 1 i then x + y 1? (X.+y.) g .. h • = 1 1 = il 1 i The maps define the Lie product y.1, j (6.11) [x,y] = l Ix. 'Yj] = l [g. ,h ] • , • ~ • • J. j 1,J 1,J We prove now that this definition does not depend on the representatives g. ,h. of the classes Xi'Yj. For if 1 J ~ = g! = x ., then g. = g!f, f E G'+ and by lemma (1.1), (ii), 1 1 1 111 l = [9i,hj]f[f,hj ]. 50 by the remarks after definition (6.8) and since [f~hj] EGi +j +l , we have 19.t.,hjJ = I91.,hjJ. 5imilarly for the second factor. TO prove the distributive laws, consider elements f,g,h E G such that f = x, g = y E grnG, fi = v E grmG. Then [x+y,v] = [fg,h] = [fg,h] = [f,h]g[g,h] = [!;h] + [g,h] = = [x,v] + [y,v]; the same argument applies to the distributivity in the second factor. We still have to prove that the new Lie product in fact ~atisfies (a) and (b) in definition (5.1). Let x be as in (6.10); then 58. IX ,xi] = Ig~ ,g.]= l = 0 in grG, and i J. J. -1 [x. ,x.] [gi,gj] = [gj ,gi] = -[xj,x ], . J. J = i so [x,x] = 0 by (6.11) • This proves (a) • Let now x,y be as in (6.10), z z., z. 17, = I• J. J. = J. J. f. E G.• In order to prove Jacobi's identity, it is enough to J. J. consider the case when x = xr ' y = Ys' Z = Zt are homogeneous. Since (~) = g for aIl g,h E G, we have by (iv) of Lemma (1.1) [[x,y],Z] + [[z,x],y] + [[y,z],x] = which is 0 in grG. The theorem is now proved. We now use thefr·amework: introduced previously for a proof of the Dollowing theorem. (6.12) Theorem (Magnus). Let F be the free group (free1y) generated by X = {xi; i El}; then DnF = rnF, n = 1,2, ••• Proof: (a) Consider the filtrations of F given by the two descending central series {DnF; n = 1,2, ••• } and {rnF; n = 1,2, ••• }. Since DnF ~ rnF for aIl n, we have a map ~n: gr~r)F + gr~D)F induced by the inclusion, which extends to a Lie algebra homomorphism ~ : gr(r)F + gr(D)F. Moreover, since F is freely generated by the elements of X, then X is also a set of generators for the 59. free abelian group grir)F = F/[F,F] = griD)F. This last equality follows from the fact that D F = FI = r F (cf. (7.1) 2 2 below) • (b) The injection ~ : X ~ gr(r)F induces a Lie l algebra homomorphism ~: (X) ~ gr(r)F, by definition of a free Lie algebra. ~ is in fact cnte, since we gave an explicit description of the generators of in terms of elements of X in Corollary (1.13). (c) We define next an injective homomorphism n : gr(D)F ~(ZF)~t.he group algebra of F over Z. Take an element 9 E DnF representing a class 9- in gr~D)F, so that v(g) ~ n, where v is the filtration induced by the series of dimension subgroups. Now by theorem (4.10) g, considered as an element of ZF, has a unique expression (6.13) ~.- (x ' -1) • •• + l zi· . · 1 1 . . . il, ••• ,in 1 il 1 • •• (xi -1) + ô +l n n with zo' .•• 'z~ l' E Z, x" x, E X and Ô + E (~F)n+l. 1 , ••. , n 1 1, n l 1 J Moreover, since g-l E (~F)n by definition of DnF we must have Zo = l, and the expression (6.13) must reduce to 9 ~ 1 + ~ z, ,(xi.-l ) . . . (xi -1) + ô +l • 1l···1n .L n n By the uniqueness of this expression, we see that the element 9 * = l z, ,(x.. -1) . . . (x, -1) depends only on the class 1l···1n 1 1 1n 60. of 9 mod Dn+1F and not on g. So we define Since g* == 9 - 1 mod (~F)n+1, we have from formu1ae (4.6) (4.8) that n(g+h) = 11 (g) + 11 (h) since gh - 1 == (g-l) + (h-l) l'nod (~F)k+1 where k ~ v(g) =v(h) ,; and a1so n ( [g ,h]) = 11 (g) n (h) - n (fi) n (g) = [n (g) ,n (h) ] , since [g,h] - 1 == (g-l) (h-1) - (h-1) (g-l) mod (~F)m+l, m ~ v([g,h]), thus proving that n is a Lie a1gebra homomorphisme It is injective since if 11(9) = 0, 9 E gr(D)F, then n g-l E (~F)n+1 or equiva1ent1y 9 E Dn+1F, i.e. 9 = l = 0 in gr(D)F. For the e1ement xi E X c:griD)F we have 11(X.) = x. - 1 since x. = l + (x.-1) is the unique expression ~ ~ ~ ~ (6.13) for xi. Next, note that the uniqueness of the expression (6.13) for aIl n, and aIl e1ements of ZF, gives an isomorphism ZF ; ~(Y), of the group a1gebra with the a1gebra of formaI series on y ='{(xi -1); i E I} (see definition (6.4»; moreover, n maps gr~D) F into ~n' hence a1so gr (D) F into ~ c ~ - ZF. (d) Consider now the composition of the three Lie a1gebra homomorphisms defined in (a), (b) and (c) 61. (O)F éll. (X) ! gr (r)F ~ gr ~ :1l(Y) L • It maps x. -+ x.-l, i € l, and so the set of free 1 1 generators of '3i~one-to-one ontè> the set of free generators of ttt (Y) , which is in fact also the set of free generators of~.r (Y), cf. Page 53. Hence, it must be injective; in particular, Wis injective. To complete the proof we use induction: By definition, rlF = 0lF = F. Let now n ~ l, and assume rn_IF = 0 n-IF; since ~ is injective, so is its restriction o IF ,10 • -+ n- 'l'n • O F = , n which is also onto since it was induced by the inclusion. Hence we must have r n F = 0 n F. (6.14) Corollary of the Proof. For a free group F on the set X of free generators Proof: We have remarked in part (b) of the proof that the map CI> : {If ~ -+ gr (rt was onto. In (d) we proved that it was injective. 62. ,§ '1'.' CASES' h' i::' '2',' 'n' i::'3' FOR AN ARB'ITRARY GROUP The analogue of Magnus theorem for an arbitrary group has not yet been proved in general. However it has been shown that for any group G, D2G = r 2G and D3G = r 3G. In this section we give a straightforward proof of the first equality, and also present the very ingenious proof of the second one, due to A.H.M. Hoare [7]. (7.l) Theorem. For any group G, D2G = r 2G. Proof: As remarked in (4.5), â = âG, the difference ideal of the group ring ZG, is a free abelian group on {g-11 9 E G}. Thus, there is a homomorphism ~ : âG ~ GIGI induced by the mapping ~(g-l) = g. The homomorphism ~ is 2 obviously onto. Moreover â 2 is mapped onto 11 if a E â , then a = ~ Z h{g-l) (h-I), Zgh E Z, so it is enough to g,hEG 9 check where does (g-l) (h-l) go. Now (g-l) (h-l) = (gh-l) - (g-l) - (h-l) {cf. (4 • 6) ) , l I so 'fi~ maps ~'t on t 0 ghg-lh- ~... G • Hence we have a surjective map On the other hand we have a map ljJ from G into 2 2 â/â given by ljJ{g) = g-l mod â • It is a homomorphism since 2 gh - 1 :: (g-l) + (h-l) mod â again by (4.6). Also I 2 ljJ a map G ~ D2G = ' {91 9 - lEâ }.' so that induces 63. ... tjJ . G/G I -+ Â/112 which is clearly onto. The kernel of tjJ is precisely D2G, ... I hence ker tjJ ;::: D2G/G • Furthermore, it is easy to see from the definitions of cp and tjJ that the compositions ...... tjJ 0 cp . 6,/6,2 -+ 6,/6,2 , ...... cp 0 tjJ G/G I -+ G/G I ... are in fact the identity; thus ~ is an isomorphism, ker tjJ ;::: 1 and we have (7.2)· The·orem. For any group G, r 3G ;::: D3G. We will need the following lemma, whose proof will be given later in this section. (7.3) Lemma. If B is a matrix over the ring Z (or more generally over a commutative Euclidean ring E), then B is antisymmetric if and only if there is a matrix Y over Z such that B = YB - BIy I , where BI stands for the transposed matrix. We know that any group G is a quotient of a free group. So let F be free on {x.; i E I} and let l. ~ 1 -+ R -+ F -+ G -+ 1 64. be exact, i.e. ~ is onto and R = ker~; also let ~* :; ZF -+ ZG " be the homomorphism induced by ~ .and R* = ker ~*. (7.4) Lemma. If ~(f) = g, f E F, g E G, then g-l E (~G)3 if and only if there is a p E R* such that f - l + P E (~F)3. 3 Proof: g-l E (~G) implies that there is an element ô E (~F)3 with ~*(ô) = (g-l) = ~*(f-l). Put p = ô - (f-l); then p E R* and f-l+p = ô E (~F)3. In order to prove theorem (7.3) it is enough to prove (7.5) Lemma. If f - l + P E (~F)3, f E F, for sorne p E R*, then there is an r E R such that fr - l E (~F)3. Assume the lemma, and let g E D3G, f E F be such that ~(f) = g. Now g E D3G if and only if .~ - l E (~G)3, and by lemma (7.4) this holds if and only if there is a p E R* with f - l + P E (~F)3. 50 (7.5) provides an r E R such that fr - l E {~F)3. But then g = ~(f) = ~(fr) E r 3 G because fr E D3F, which by Magnus' theorem is equal to r 3 F. proof·o·f Temma (7.5). Let p E R*. By lemma (4.12) we have 65. Cr) p = . [,~ v Cr-l) , ver) e ZF, so reR (7.6) v (r) = z (r) + ô (r) , z (r) e: Z, ô (r) e: ÂF. z (r) Let s = IIr ; is zero exeept for a finite number of rIs, henee the produet is finite and is taken in any order. (r) Then s e: Rand by (4.6) and (4.7) IIrz - l :: 1: z (r) (r-l) mod (ÂF)2. Thus s - l - 1: z(r) (r-l) e: R* n (ÂF)2, and it has r an expression 1: y(r) (r-l) with y(r) e: ÂF. We obtain re:R (7.7) s - l = 1: z (r) (r-l) + 1: y (r) (r-l) , y(r) e: ÂF. re:R re:R Let p,f be as in the staternent of the lemma, let cr = p - (s -1) - ( f -1) (s -1) and b = f s • Then by (7. 6 ) and (7. 7) cr = - (f-l) (5-1) + 1: v (r) (r-l) - 1: z (r) (r-l) - 1: y (r) (r-l) r r r (7.8) = - (f-l) (s-l) + 1: (ô (r) _ y (r) ) (r-l) e: (ÂF) R* , r and sinee cr = p - (s-l) - (b-f-s+l) = p - (s-l) - (b-l) + (f-l) + (s-l) we have that (7.9) (b-l) + cr = (f-l) + P e: (ÂF) 3 by the hypothesis in the lemma. Thus cr and (b-l) + cr belong to (ÂF)2, henee also (7.10 ) b - l e: (ÂF)2 \. 66. and by Magnus' theorem b E f 2F, so (7.11) (b-1):: 1: b .. (x.-1) (x.-1) mod (llF)3 i,jEI 1J 1 J where b .. -b .. E Z and x. ,x. be10ng to the I} 1J = J1 1 J set "{xi; i E of generators of F. This fo110ws from the fact that b .. 1J b :: II [x.,x.] mod f3F (cf. (1.13», and that by (4.8) 1,J. . 1 J [x. ,x.] 1 :: (x.-1) (x.-1) (x. -1) (x. -1) mod (llF) 3 • 1 J - 1 J - J 1 Now by (7.8) cr E (llF)R*, therefore it can be written as cr (r) cr = 1: cr (r) (r-1) , E llF . r Let cr (r) - 1: w~r) (x. -1) mod (llF) 2, w~r) E Z i 1 1 1 and let (7.12 ) r - 1 - 1: r. (x .-1) rnod (llF) 2, r. E Z , j J J J so that (7.13) cr _ 1: 1: w ~ r) r . (x. -1) (x. -1) rnod (llF) 3 • . . 1 J 1 J r 1,J We had (b-1) + cr E (llF)3 in (7.9), or equiva1ent1y (7.14 ) b - 1 :: -cr rnod (llF)3 ; consequent1y (7.11), (7.13) and uniqueness in (4.9) give (7.15) -b .. = 1: w ~r) r. = b ... 1J rER 1 J J1 Let B be the rnatrix with entry b .. in the ij-th 1J place. From the 1emma (7.3), there is a rnatrix Y' (y .. ), = 1J 67. e y .. E Z such that ~J (7.16) b .. = 2 (Yikbkj - bk;Yjk) • ~J k Let (r) (7.17) R,i (r) = 2 YikWk E Z, k R,i(r) (7.18) fer) = II x. E F, i J. and t = II [fer) ,r], where the finite products over i and r r are taken in any order. R is normal in F, so that tER. By (7.12), (7.18) and (4.8), modulo (L\F)3 [f (r) :,r] - 1 _ (f (r) -1) (r-1) - (r-1) (f (r) -1) - 0: R, ~r) (x, -1» (2 r, (x .-1» i J. ~ j J J - 0: r. (x.-1» (2 R,~r) (x,-l» i ~ J. j J J 2 (R,~r)r. r,R" (r) ) (x.-1) (x.-1). • , J. J J. J ~ J ~,J Therefore, a1so mod (L\F)3 t - 1 = 2 ([f(r) ,r] - 1) = L (2 (R,~r)r,- r.R,~r» (x.-1) (x.-1). .. J. J ~ J ~ J r J.,J r But by (7.17), (7.15) and (7.16) 2 (R, ~r) r ,- r R, (r» = r J. J i j. = ~ (Yik(-bkj ) - Yjk(-bki» = -b. ,= L W ~r) r .• ~J rER ~ J 68. Hence by the above and (7.11), a1ways mod (ÂF)3 t - 1 - 1: ( l w~r)r.) (x.-1) (x.-1) - cr i,j rER 1 J 1 J and since b - 1 E (ÂF)2 (bt-1) = (b-1) + (t-1) + (b-1) (t-1) :: (b-1) + (t-1) • From the 1ast two congruences and (7.14) we arrive at (bt-1) :: 0 mod (ÂF)3, that is to say f(st·)~l E (ÂF)3, where st E R. The proof of 1emma (7.5) is now complete. Proofof 1emma (7.3). Let B be a matrix with coefficients in Z. If B = YB - B'Y', it is c1ear1y antisymmetric. For the converse we use induction on the order n of B. The 1emma is obvious1y true for n = 1. So let n > 11 we can assume that the highest common factor of the e1ements of B is the identity. Hence there are e1ements zik E Z such that (7.19) 1,2, ••• ,n Let r kj = b ij , r'sj = 0 if s :f k, j = and let (r.. ). Then sj 69. bll ... blk • • • blm 0 . .. 0 . • • · Bl\R. = . • · · = (bikb R.j) bR.l bR.n· b b bn1 ... nk . .. nm ·• 0· 0 Let (8.20) so that We have s .. -Sji and sik = 0 Skj aIl i and j. If we ~J = = delete from S the k-th row and column we obtain an (n-l) x (n-l) antisymmetric matrix Sk. By induction hypothesis, there is a matrix Tk such that , k=1,2, ••• ,n. Now we add to Tk a row and column of zeros to form the k-th row and column of a new n x fi matrix T. From this we have that (8.21) S = TS - S'T'= ST' - TS' since S is antisymmetric. Put then (8.20) (8.22) = (b. kb n' - b. nbk' - b nkb .. ) ;::: S 1. JV J 1. JV J ~. JV 1. J If rrow Ytk = RkR. - PT', YR.k = RkR. - TP', we obtain by using (8.20), (8.21) and (8.22) 70. YR,kB - B'YR,k ;:: BYR,k - YR,k B' = BRkR, - BPT' - RkR,B' + TP'B' = BRkR, - R'kR, B' - ST' + TS' = S + bR,kB - S = bR,kB• Finally, let Y = L zR,kYR,k' zR,k as in (7.19), and compute R"k Thus, lemma (7.3) is proved. 71. Although the two results we present in this section are concerned with finite groups, they are of quite different nature. The first one, due to J. Buckley [8] says that for a finite group the intersection of the groups in the lower central series and the intersection of the dimension subgroups coincide. The other, obtained by I.B.S. Passi [11] gives a hint as to where one should look for counterexamples to the generalization of Magnus' theorem to arbitrary groups, if indeed such examples existe In the proof of the first result, we will need the fOllowing characterization of finite p-groups. (8.1) Theorem. Let G be a finite group. Then G is a p-group if and only if n (8G)k = o. k Proof: (a) If n (8G)k = 0, then n DkG = 1 and k k since rkG ~ DkG for all k, also n r k G - 1. But G is k sorne integer n > 1. finite, so we must have r n G = 1 for Hence G is nilpotent. We know that a finite nilpotent group is the direct product of its Sylow subgroups, In order to prove that there is only one of them, we sho\'l that if 9 and f are elements of G that commute and have different prime power orders, then n (8G)k ~ o. k 72. Let f be of order p,g~.;of order q, (p ,q). = 1 and fg = gf. Put l 2 f ' = f P- + f P- + ••• +l' ,g = 9q~1 + 9 q-2 + ••• an easy computation shows (.g-l) g' = 0, (g-l) f' = o. Let y,z € Z be such that yp + zq = 1 and let cr = yf' + zg'. Then (8.2) (f-l) (g-l) cr = y(f-l)f' (g-l) + z(f-l) (g-l)g'= 0 since f and 9 conunute. Co:nsider the image of cr under the augmentation homomorphism (4.3). lt is equal to y € (fi) + z € (g') = yp + zq = 1; hence cr-l € ÔG. Given that (f-l) (g-l) € (ÔG)2 and that by (8.2) (f-l) (g-l) (cr-l) = -(f-l) (g-l) we may conclude that (f-l) (g-l) € (ÔG)k for aIl K. (b) Assume now that G is a finite p-group, say IGI = p~ , p a prime, ~ ~ 1. Let KG be the group algebra of G over the field K = GF(p) with p elements, i.e. KG is the vector space over K with (finite)basis {g; 9 € G} with multiplication induced by the product in G. Let l be the augmentation ideal of KG. The elements g-l, 9 € G, forro a basis over K for the vector space l. Now we show that l is a nilpotent ideal. We will need the following result Let G be a p-group and K a finite field of characteristic p. The only irreducible finite dimensional KG-module is the trivial module • 73. Proof: Let M be an irreducible KG-module i s ince '1 K1 = P 1T 1T > 1, we have IMI = p1T~ 1T ~ 1, where m = dim K M. Let us consider G as a permutation group acting on M* , the set of elements of M, and let Fixg M* be the set of elements of M left fix by each g E G. Since G is a p-group, IGI = pcc, cc ~ 1, every orbit X of M* has Ixi = pn for sorne n > 0, with n > 0 unless X consists of a fix point. Now, M is the dis- joint union of orbits with respect to G, so we have 1FixGM* 1 - 1M*I ~,"1od P, hence 1 FixGM* 1 - 1M 1 mod p, hence 1FixGM*1 - 0 mod p. Since G fixes'{O}, there is at least one more fix point. This point generates a trivial sub-module, which must be aIl of M because M is irreducible. By what we have just proved, if M is a finite dimensional irreducible KG-module, one has gm = m for aIl g E G, aIl m E M, or (g-l)m = 0 , i.e. lM = o. Let N be a finite dimensional KG-module and N = N ,N , ••• , 1 2 Nn- l = {a} a composition series for N. The composition factors N. II N. are irreducible KG-modules, and by the above 1.- 1. I(N. l/N~) = 0 , IN. 1 < N., i = 2,3, ••• , n 1.- 1. 1.- - 1. Hence, l n-l N=O. For N=I it follows that l n =0, .n>'o " and so l is nilpotent. 74. The canonical homomorphism ~ : ZG + KG, ~(g) = g, for g E G and ~(z) = Z (mod p), maps ~G onto l and (~G)n cnte n l =0. If ô = ! Zgg E (~G)n, we must have Zg :: 0 mod p since 1T (ô) = o = 1T (z ) 9 and {g;g E G} is the basis for KG over K. l 9 Thus (~G)n€ pZG and consequently n (~G) k ~QpkZG = O. (8.3) Lemma. Let N be a normal subgroup of G. N has prime power index in G if and only if l (N) = 11 (I (N) + (~G) k) • k=l Here I(N) is the ideal of ZG defined in lemma (4.12). Proof: If we identify Z(G/N) with ZG/I(N), then (~.~/N»k corresponds to «~G)k+I(N»/I(N). Now by theorem (8.1), G/N has prime power order if and only if fl (~(G/N»k= O,i.e. if and only if k «~G)k + I(N» = I(N). k (8.4) Theorem: Let G be a finite group. Then 00 = ('\ k=l Proof: Clearly n r kG ~ " DkG since r kG ~ DkG, k = 1,2,... Now let 9 E n DkG so that g-l E () (~G)k. k k BY the lemma (8.3), g-l E I(N) for all normal subgroups . of prime power index in Gr and by lemma (4.12) 9 E N. Therefore, 9 E n rkG by theorem (2.10). And the proof is complete. k 7S. (8.S) Theorem. If G is a group such that r n G ~ D n G for sorne integer ne> 3), then there exists a finite p-group L for which also rnL ~ DnL. Proof: Suppose such a Gand n exist, that is, there is an element 9 E G such that 9 E DnG but 9 ~ rnG. Then g-l E (~G)n, and therefore g-l is a finite linear combination with integer coefficients of products (xp -1) (xp -1) ••• (xp -1), x E G. Let H be the 1 2 n Pi . subgroup of G generated by the x p 's. Then B is finitely l. generated and (8.6) E D H, but 9 n but ~ r n H since r n H -< rnG. Next, put K = H/rnH; K is a finitely generated nilpotent group and r n K = 1. Let g' denote the coset of 9 in K. We have g' ~ 1 by (8.6), and so g' ~ rnK = 1; but g' E DnK since DnK is the image of DnH under the homomorphism induced by the canonical map from H onto K. We know that finitely generated nilpotent groups satisfy the maximal condition (theorems (1.24) and (2.4) ). Therefore, we can find a normal subgroup M maximal with respect to the property that g' ~ M. 76. Again, put L = K/M and let f be the eoset of g' in L. Sinee g' t M, f is not the identity. Moreover, rnL = 1 sinee rnK = l, so that f t rnLi and f € DnL by the same argument as before. Now let us study L. It is finitely generated and nilpotent. Moreover, if N is a non-trivial normal sub- group of L, then f € N. Otherwise, let NI be a normal subgroup of K such that N = NI/M. If f t N, then g' t NI. But NI > M sinee N ~ l, eontradieting the maximality of M. - Henee L is sueh that the element f ~ 1 belongs to aIl its normal subgroups. By the eorollary (2.8) to Hirseh's Theorem, we eonelude that L is a finite p-group. 77 • REFERENCES [1] Marshall HALL, Jr., The Theory of Groups, The Macmillan Company, New York, 1959. [2] Philip HALL, Nilpotent groups, Lecture Notes Canadian Math. Congr., Summer Serninar 1957. [3] W. MAGNUS, A. KARRASS, D. SOLITAR, Cornbinatorial group theory, Interscience Publishers, 1966. [4] N. JACOBSON, Lie algebras, Interscience Publishers, 1962. [5] J~P. SERRE, Lie algebras and Lie groups, 1964, Lectures given at Harvard University, Benjamin, New York, 1965. [6] P.M. COHN, Generalization of a theorern of Magnus, Proc. London Math. Soc. (3), 2(1952), 297-310. (See also corrigenda in the same volume). [7] A.H.M. HOARE, Group rings and lower central series, J. London Math. Soc. (2), 1(1969), 37-40. [8] J. BUCKLEY, On the D-series of a finite group, Proc. Amer. Math. Soc., 18(1967), 185-186. [9] J.P. LABUTE, On the descending central series of groups with a single defining relation, Journ. of Algebra, 14 (1970), 16-23. [10] K.W. GRUENBERG, Residual properties of infinite soluble groups, Proc. London Math. Soc., 7 (1957),29-62. 78. [11] J.B.S. PASSI, Dimension subgroups, Jour. of A1gebra " 9 (1968),152-182 • •• [12] W. MAGNUS, Uber Beziehungen zwischen hoheren Kommutatoren, Jour. fur•• Math., 177(1937), 105-115.