Arch. Math., Vol. 44, 488-492 (1985) 0003-889 X/85/4406-0488 $'2.50/0 �9 1985 Birkh~iuser Verlag, Basel

Locally nilpotent groups with an intersection property

By

C. J. B. BROOKES*) and H. HEINEKEN

1. Introduction. Recent attempts [1, 2, 3] to characterise the group algebras k G that are primitive have established the importance of group propert ies of the type: given a subgroup B of G, all abelian subgroups of G of infinite rank intersect B non-trivially. For example, Theorem D of Brookes-Brown [2] says that for a non-trivial countable nilpotent group G with centre Z, and a field k, the algebra k G is pr imit ive if and only if k is countable, G is torsion-free and there exists an abel ian subgroup A of infinite rank with A n Z = 1. In this paper we let B be a subgroup of a term of the upper central series of G and consider the effect of the imposi t ion of the above property. Note that if B is trivial we are just saying that all abel ian subgroups are of finite rank, a property already well-documented (see [4]).

We shall denote the upper and lower central series of G by (, (G) (i --- 0) and 7, (G) (t >-- 1) respectively. G is said to have finite upper central height if its hypercentre is (s (G) for some natural number s.

The groups considered by Brookes in [1] had no non-trivial per iodic normal subgroups and thus had torsion-free hypercentres.

Theorem A. Let G be a group with torsion-free hypercentre and B be a subgroup of (r (G) for some r ~ N. Suppose all abelian subgroups of G having trivial intersection with B are offinite rank. Then G hasfinite upper central height.

This has the immediate consequence.

Corollary A 1. The only torsion~,ee hypercentral groups G with subgroups B as in the theorem, are nilpotent.

Let H be a subgroup of G. Then B n H_-< ( r ( H ) if B _-__ (r(G). Also, if A is a subgroup of H with A ~ ( B c ~ H ) = I then A c ~ B = I . Thus the theorem can be applied to all subgroups of a torsion-free group to give

Corollary A2. With G and B as in the theorem, suppose further that G is torsion-free. Then all subgroups of G have finite upper central height.

We can indeed do better than Corollary A 1.

*) The first author would like to thank the Alexander von Humboldt Foundation and the University of Wiirzburg for their support and hospitality.

Vol. 44, 1985 Locally nilpotent groups 489

Theorem B. Let G be a torsion-free locally nilpotent group. Suppose B <-(r(G)for some natural number r and all abelian subgroups having trivial intersection with B are of ]inite rank. Then G is nilpotent.

By putting B = 1 here we can retrieve that torsion-free locally nilpotent groups of finite abelian subgroup rank are nilpotent, a theorem of Mal'cev [5].

As observed in [2], in the free nilpotent group of class 2 on an infinite generating set every abelian subgroup having trivial intersection with the centre is cyclic. On the other hand for any class greater than 2 such a free nilpotent group contains an abelian subgroup of infinite rank which meets the centre in 1. In general if A is an abelian group of infinite rank intersecting trivially with (r(G) where G is a free nilpotent group of class n, then for some s<=(n- r) it contains an infinite subset of ys(G), linearly independent modulo Y~+l (G). But Moran [6, Theorem 1.3] says that such a subset freely generates a subgroup of G nilpotency class [n/s]. So A can only exist if s > �89 n and thus only if n > 2 r. In particular, any free nilpotent group of class 2 r has the property that all abelian subgroups intersecting trivially with (r (G) are of finite rank. On the other hand we prove

Theorem C. Let G be a torsion-free nilpotent group with subgroup B <-~r(G). Suppose that all abelian subgroups of infinite rank intersect non-trivially with B. Then there is a normal subgroup GI of G of nilpotency class at most 2 r, with G/G1 torsion-free of finite rank.

Theorem B can be used to tidy up Theorem C of [2]. As noted in [2], to deal with the implication still requiring proof it may now be claimed that the group is nilpotent and Theorem D of [2] can be appealed to. We say a subgroup H of G is dense if some positive power of every element of G lies in H. Let ~3 denote the class of torsion-free locally nilpotent groups whose finite rank subgroups are finitely generated. Countable locally nilpotent groups contain dense ~-subgroups [2, Lemma 5.1 (ii)]. Our primitiv- ity result now reads

Theorem D. Let G be a countable torsion-free hyperabelian locally nilpotent group and let k be a field. Let Z be the centre of G. (i) Suppose k is countable. Let H be a dense ~3-subgroup of G and Zo = H ~ Z. Then the following statements are equivalent.

(a) kG is primitive. (b) There exists an abelian subgroup A of G of infinite rank with A c~ Z = 1. (c) kH(kZo) -l is not Noetherian.

(ii) Suppose k is uncountable. Then k G is primitive !land only (f Z = 1.

2. Proof of Theorem A. We shall need

Lemma 1. Let G be a group with torsion-free hypereentre K. Then K/~r(G) is torsion- free for all r ~ IN.

P r o o f. It is enough to show that K/~I (G) is torsion-free to provide the basis of a simple inductive argument. Suppose g ~ K with 9 n central in G. Certainly 9 is central an K, as in a torsion-free hypercentral group the centre is isolated. Consider the action

490 C . J . B . BROOKES a n d H. HEINEKEN ARCH. MATH.

a n a n of G on the torsion-free abelian group A = (1 (K). If a ~ A then [ , gl] = [ , gl] for g~ ~ G. So [g, gl] ~ = 1 for all gl E G. The torsion-freeness of A shows [g, gl] = 1 and so g is central.

In proving Theorem A we may suppose B = (~(G). Because [ _ ~ t ( G ) , y t ( G ) ] = 1 the subgroup At = ~t(G) ~ 7t(G) is abelian. Moreover At centralises every A~, u => t, since A~ <= ?t(G) and At <= ~t(G). Let A = ~--~ At. The supposit ion on abelian subgroups with

t

trivial intersection with B ensures AB/B is of finite torsion-free rank. Let K be the hypercentre of G. By Lemma 1 K/(r(G) is torsion-free and so its subgroup AB/B is too. AB is a product of normal subgroups of G and therefore is also normal. It lies in K. Thus AB/B is G-hypercentral and torsion-free of finite rank. Hence it is G-nil- potent [7, Lemma6.37]. In other words A B = A ~ ( G ) < = (~(G) for some v. Hence At --- A =< ~'~ (G) for all t. In particular A~ + l = 7~ + 1 (G) c~ ~ + 1 (G) is in ~'~ (G). So the G- hypercentre of ?~+1 (G) is ?~+1 (G) ca (~(G) and is G-nilpotent. That is, K m ?~+1 (G) is G-nilpotent (of at most class v). But K/(K m ),~+~ (G)) is G-nilpotent (of at most class v). So K is also G-nilpotent. Hence the hypercentre of G is (s (G) for some s -< 2 v.

3. Proof of Theorem B. Suppose G is non-nilpotent. Taking a countable non- nilpotent subgroup H of G we know B c~ H ~ ( r (H) and all abelian subgroups of H having trivial intersection with B c~ H are of finite rank. Thus by renaming H as G we may assume G to be countable.

By Theorem A the hypercentre K of G is (s (G) for some s ~ N. Because K => B all subgroups of G having trivial intersection with K are of finite rank. Note that G/K is torsion-free, because G is. We shall show that G/K, which is non-trivial because G is supposed not to be nilpotent, has a non-trivial centre. This is a contradiction of the definition of K and so G must be nilpotent after all.

Since G is countable G = U N, where N, =< N I if i < j and K =< N~ and Nz/K is !

finitely generated (i ~ N). Local nilpotence of G implies that each Ni is nilpo- tent, remembering K = (s(G). Let A, = ?s+J (N~) c~ (s+l (N,). It is abelian because [Ts+l (N,), ~s+l (N,)] = 1. Also [A, AI]= 1 ( i < j ) because Ai <= ?~+l (Ni) ~ );s+l (Nj) and Aj_-<(,.+I (N/). So A = ~ A , is abelian. Because all subgroups having trivial intersec-

t tion with K are of finite rank, AK/K is of finite torsion-free rank. But as already noted G/K is torsion-free and so AK/K is torsion-free of finite rank. Let F be a subgroup of AK containing K such that F/K is free abelian of finite rank and A K / F is periodic. Because F/K satisfies the maximum condition on subgroups the ascending sequence { 1-[ [(A~mF) K]} s t o p s a t j = t , say. l < t ~ /

Write -'s for images in G/K. Because K =< F we have (A, c~ F ) = A, m P for each i. So we know

(1) ~c~F<= ~ (3,•F) l<=t<_t

for all j. Note that Fc~?~+l(G)=U(Fc~?s+l(N,)) where {Fc~?~+I(N,)} is an I

Vol. 44, 1985 Locally nilpotent groups 491

ascending sequence of subgroups of ft. This sequence must become stationary and so E = F c~ 7s+1 (G) = F • 7s+ i (Nk) for some k that may be taken as at least t, and

(2) F c~ ~+ 1 (AT,) = /~ (i -->_ k) .

Let j>k>=t . Then ~+jc~F=)~s+~(N/+l)C~(s+l(Nj+l)C~ff. This by (2) is E c5 ~s+l (NI+I)- So

(3) Aj+I ~ F = F ~ Ys+l(Nj) 0 ( s + l ( N l + l ) .

But also from (1) we know (Aj+I c~ F) =< Nt -< 37j. So

Aj+I c~ ff _-< F c~ Ys+I (Nj) n (Nj c~ (s+l (Nj+I)).

But Ni~(~+l(N/+i)<=(s+l(Nfl and so A/+Ic~F<=F~AI. So ,4kc~ff>Ak+l~P>-- . . . is a decreasing sequence inside F, a group of finite torsion-free rank.

Next we show each factor ( ~ c~ F)/(AI+ 1 c~ if) is torsion-free. This is because from

(3) it is (F c~ ~s+~ (Nil ~ ~+~ (Nj))/(F c~ 7~+~ (Nil c~ (s+l (N~+l)). This is isomorphic to

(F c~ ~.+~ (N~) c~ ~s+l (Nil) ~s+~ (Ns+l)/~+~ (Ns+l), a subgroup of Ns+~/~s+~ (Ns+i). The latter is isomorphic to Nj+i/(s+~(Ni+i) because K=(s(G)<=(~+I(Ns+~). Since Nj+l is torsion-free Nj+~/(~+~ (Ns+~) is.

So Am ~ F=Am+l ~ / ~ = : . . for some m ~ N. Let ~2=A m c~ ft. We claim it is non- trivial. To see this note that if At is trivial then Ys+* (Ni) --< (~ (N,) and so N, is of class at most 2 s + 1. If N~ is of class at most 2 s + 1 for all i > m then G is nilpotent of class at most 2 s + 1, a contradiction. So A, is non-trivial for some n => m. It is torsion-free because (~ is. But A,/(A, c~ if) is periodic and so A, c~ ff is non-trivial. But A , ~ F = C .

Finally we observe that C ~ A/ (i _-> m) and so C -< (,+ I (N,) (i_->m). Hence C = < (~+l (G). Thus ~'~+~ (G) >- C > K - (s(G), a contradiction o f K being the hyper- centre of G.

4. Proof of Theorem C. We shall need

Lemma 2. Let H be a torsion-free nilpotent group of class r + t + 1, with r, t ~ N, such that 7r+l (H) (t (H)/(t (H) is torsion-free of finite rank. Then H has a characteristic subgroup C of class at most r + t with H / C torsion-free of finite rank.

P r o o f. There is a free subgroup U (t (H)/(t (H) of ~r+ 1 (H) (t (H)/(t (H) of the same finite rank, freely generated by a finite set of elements of the form [[sl,s2] . . . . . Sr+l] ( t(H). The set J of elements s, obtained in this way is certainly finite. Let h be any element of H and v i s J ( 1 _-__ i - r ) . The mapping h [[h, Vl], ..., vr] ( t (H) is a homomorphism of H into )~r+l (H) ( t (H)/ ( t (H) . Let K be its kernel. Thus H/K is abelian of finite rank.

Consider L =(h ~ H:[[h, vl] . . . . . v~] s ( l (H) for all choices of v~ . . . . . vr from J > . Since there are only finitely many elements in J , there are only finitely many collec- tions Vl, ..., v~ and L is the intersection of finitely many groups with the same method of construction as K. So L is the intersection of finitely many normal subgroups with quotient group abelian of finite rank. So H / L is also abelian of finite rank. We shall now show that [Yr+ l (H), L] =< ~t- t (H).

492 C.J.B. BROOKES and H. HEINEKEN ARCH. MATH.

Clearly [7,.+1 (H), L] -< 7r+2 (H) - ~t(H). Let u~ (t(H) = [[Sl, $2] . . . . . Sr+l] (t(H), one of the generators of U (t (H)/(t (H). Choose any element x of L. Then

(4) [x, u,] - [x, [[sl, s2] . . . . . S~+l]] = 1-~ [Ix, Sn(l)] . . . . . s~(~+1)] ~ mod ( t - i (H) 7[~ Sr+ 1

where S,.+~ is the symmetr ic group on (1 . . . . . r + 1} and e ~ the signature of an ele- ment ~. The latter equivalence in (4) is produced by repeated use o f Hall 's com- mutator formula. Because x is in L the commuta tor [Ix, s~l)] . . . . , S~ ) ] is in ( t ( H ) and hence from (4) [x, u,] ~ ( t - t (H). So [U, L] < ( t - i (H). But H is torsion-free and so H/(t- i (H) is, and in the latter centralisers are isolated. Hence [7,.+ i (H), L] -< ( t - 1 (H).

Let C= CH(7,.+1 (H) (t-1 (H)/(t-I (H)). Thus C is character is t ic in H and contains L. So H/C is abel ian of finite rank. It is torsion-free since central isers in the torsion- free group H/(r- 1 (H) are isolated. The class of C is at most r + t.

To prove Theorem C we may assume B = (r (G). Let A be a ma x ima l abe l i an normal subgroup of G. Because G is torsion-free, G/B is too. The suppos i t ion ensures that AB/B is of finite rank. In a ni lpotent group a max ima l abe l ian normal subgroup A satisfies A = Ca(A). Since G is torsion-free the central iser of any subgroup is isolated and so our A is.

Let Go = C6(AB/B). Then G/Go is a unipotent a u t o m o r p h i s m group of the finite dimensional Q-vector space (AB/B)| So G/Go is tors ion-free of finite rank. Because A is contained in (,.+ 1 (Go) it follows that 7~+ i (Go) -<- Ca (A). But the lat ter is A. So the chain 1 <- B <- AB <- Go demonstra tes the n i lpotency class of Go to be at most 2 r + l . But B<(,.(Go) and so ?~+l(Go)(~(Go)/(~(Go) is an image of 7~+1 (Go) B/B, a subgroup of AB/B. So ~+1 (Go) (r(Go)/(~(Go) is o f finite rank. It inherits torsion-freeness from Go. So L e m m a 2 may be app l i ed and so Go has a characteristic subgroup G 1 of class at most 2 r, with Go~G1 tors ion-free of finite rank. Thus Gl is normal in G and G/GI is torsion-free of f inite rank.

References

[1] C. J. B. BROOKES, The primitivity of group rings of soluble groups with trivial periodic radical. To appear in J. London Math. Soc,

[2] C. J. B. BROOKES and F,L A. BROWN, Primitive group rings and Noetherian rings of quotients. Preprint.

[3] K. A. BROWN, Primitive group rings of soluble groups. Arch. Math. 36,404- 413 (1981). [4] P. HALL, The Edmonton notes on nilpotent groups. London 1969. [5] A. I. MAL'CEV, On certain classes of infinite soluble groups. Mat. Sb. 28, 567-588 (1951) -

Amer. Math. Soc. Transl. (2) 2, 1-21 (1956). [6] S. MORAN, A subgroup theorem for free nilpotent groups. Trans. Amer. Math. Soc. 103,

495-515 (1962). [7] D.J.S. ROBINSON, Finiteness conditions and generahsed soluble groups, Vol. II. Berlin 1972.

Anschrift der Autoren: C. J. B. Brookes Corpus Christi College Cambridge Great Britain

Emgegangen am21.7.1984

H. Hemeken Mathematisches Institut der Universit~it Wtirzburg Am Hubland D-8700 Wfirzburg