Bibliography. UNRESTRICTED FREE PRODUCTS, AND VARIETIES OF TOPOLOGICAL GROUPS

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1 A DUALITY IN ABSTRACT ALGEBRA. 73 Bibliography. 1. Birkhoff, G., "On the structure of abstract algebras", Proc. Cambridge Phil. Soc., 31 (1935), Maclane, S., " Groups, categories, and duality ", Proc. Nat. Acad. Sci. U.S.A., 34 (1948), 263. The University, Manchester. UNRESTRICTED FREE PRODUCTS, AND VARIETIES OF TOPOLOGICAL GROUPS GRAHAM HIGMAN*. 1. Introduction. Let O a, aea, be an infinite set of groups. It is well known that there exist two direct products of this set of groups, the restricted direct product and the unrestricted direct product. It is evident also that the free product of the groups is more nearly related to the restricted than to the unrestricted direct product. The first object of this note is to define an unrestricted free product of the set of groups, which shall have a relation to their ordinary free product similar to that of the unrestricted direct product to the restricted direct product. This we do in 2. ' The rest of the paper concerns the special case in which the O a are a countable set of free cyclic groups. Some algebraic properties of their unrestricted free product F are obtained in 3. F is defined as an inverse limit, and so has a natural topology, which is discussed in 4. The theorems there proved lead us to introduce in 5 the concept of a variety of topological groups, which is analogous to a variety of abstract algebras, in the sense of P. Hall. Finally, in 6 we discuss a subgroup of F which, without being free, has no subgroups, other than free cyclic subgroups, which are freely irreducible. The question whether such groups exist was raised by Kurosch in [4], and solved by him in [5]. The results of 6 suggest that the class of such groups may be quite extensive. 2. Unrestricted free products. To make clear the analogies, we first put into a possibly unfamiliar form the definitions of the restricted and unrestricted direct products of the groups G a, aea. Let 9t be the set of finite subsets of A, and if t esr let D r be the direct product of the groups O a for a et. Since t is a finite set there is no ambiguity in this definition. The groups D x form a directed setf if we order them by setting D x < D* * Received 17 January, 1951; read 18 January, f For directed sets of groups and their limits see Weil [11].

2 74 G. HlOMAK whenever rc«. If rc$ there is a natural isomorphism of D x into D 6, and a natural homomorphism of D* onto D x. We may, in fact, think of D$ as the direct product of D t and A, where t is the complement of r in 3, and we therefore write an element of D$ as {g, h), where g zd x and hed t. Then the isomorphism i$ x of D x into D& maps gr on (gr, 1); and the homomorphism 7r,j of Dt onto Z) r maps {g, h) on gr. Under the isomorphisms ta r the groups D x form a direct system, whose direct limit is the restricted direct product of the groups O a. Under the homomorphisms TT^, they form an inverse system whose inverse limit is the unrestricted direct produot. The corresponding procedure for free products is now clear. If r e 01, we consider now not the direct product but the free product F x of the groups Q a, aev. If rc$, F x is a subgroup of F e, and as isomorphism i 6x of F x into Ft we have the identity mapping. But if t is the complement of r in t, F$ is the free product of F x and.ft, and there is a homomorphism n ti of Ft onto F x which maps F x according to the identity mapping and.ft onto 1. Under the isomorphisms i 6t the groups F x form a direct system, whose direct limit is the ordinary free product of the groups O a, aza. Under the homomorphisms 7r vs they form an inverse system, and it is the inverse limit of this system that we shall call the unrestricted free product of the groups O at as A. Thus we formulate the following definition. An element g of the unrestricted free product of the groups O a is a set of elements gr r> one from each group F x, satisfying ^r = T rt^ whenever rc6. If g is the set of elements g x and h is the set of elements h x, then the product gh is the set of elements g x h x. It is clear that to specify an element of the unrestricted free product it is sufficient to give the elements g x for a set of values of t such that every element of 01 is a subset of at least one of them. It may be remarked that if instead of groups O a we have abstract algebras, then in general the choice of definitions disappears, and it is possible to define only the restricted free product and the unrestricted direct product. The choice in each case springs from the fact that a group necessarily possesses a unique one-element subalgebra. 3. Case of a countable set of free cycles. We consider now the unrestricted free product of O v O 2, O 3,..., where Of is the free cyclic group generated by x {. It is convenient to change somewhat the notations of the previous paragraph. The free product of the groups O v O 2,..., O n will be called J^n), and the ordinary free product of all the Q { will be called JF<">. The unrestricted free product will be called F. An element g of F determines, and is determined by, its components (f> n \ n = 1, 2, 3,...; (fri is an element of F (n), and if m < n, g^m) is obtained from tf n) by putting o^=l, i = m+l,..., n. The mapping g-+tf n) is a homomorphism of F

3 UNRESTRICTED FREE PRODUCTS. 75 onto JT (n), and if tf n) 1 for all n, then g \. It is convenient to regard the groups F w, F M as subgroups of F, by identifying z { with the element g of F satisfying cf n) = x { for all n greater than i. LEMMA 1. // g v g 2,... are elements of F satisfying $ n) = 1 if n<i, then there exist elements h 0, h v h 2,...of F satisfying an arbitrary set of relations of the form ^,_ 1 = w { {g i} h { ), i = 1, 2,... We set ^[ n) = 1 if i >«., and determine &j. n) for i = n 1, n 2,..., 0 successively from the equations (A) fcft-uvfoj'w). We note that this equation holds for all values of *; for * < n we have made it a definition, and for i > n both sides are 1. Now the homomorphism of.f (n) onto JP (m), m<n, which maps x { on 1, i = m-\-l,..., n, maps h\ n) on A m). Ifi^n this is true because both these elements are then 1, and it can be extended successively to smaller values of i by equations (A), and the corresponding equations with n replaced by m. Thus for fixed i the elements h[ n) are the components of an element h { of F. Since equations (A) hold for all n, we havefe,-^ = w i (g iy h ( ) as required. LEMMA 2. Ifg v g 2,... are elements of a free group 0 no one of which is 1, then there exist integers X v A 2,... such that the equations h^ = g ( fyi do not hold simultaneously for any elements h 0, h v h 2,... of 0. If l{g) denotes the length of the element g of 0 with respect to a fixed generator system, we take A, = l{g t )-\-2. Suppose that elements h 0, h v... do exist in 0 and satisfy the equations of the lemma. If h^l, then It follows that This implies that h { _ x ^ 1, and we may repeat the argument, obtaining by induction l(h { _ r ) ^l(h t )-{-r. This clearly implies that if i^l(h 0 ) then h { = 1. But then also if i ^ l(h n )-\-l then & = 1, contrary to assumption. The lemma follows. It follows immediately from Lemmas 1 and 2 that F is not a free group. But the following theorem is stronger. THEOREM 1. If y is a homomorphism of F into a free group, thm for torn* integer n, ^n) = 1 implies y(y)=l. If there is no such integer, choose for each integer n an element g n auoh that g< n) = 1, but y(g n ) = 1. By Lemma 1, whatever the integers X lt A 8,... we can find elements h 0, h v... to satisfy h i _ 1 = g i fyi. The images y(g { ) and y(hf) must satisfy the same relations, but, by Lemma 2, if A l5 A a,... are properly chosen this is impossible.

4 76 G. HIOMAN COROLLARY. A free group which is a homomorphic image of F is finitely generated. 4. The topology of F. The definition of F as an inverse limit carries with it a topcflogy. If X is a subset of F, the element g belongs to the closure of X if given n we can find x in X such that g^n) = a/ n). Or we may say that the kernels of the homomorphisms g->(f n) are a basis of neighbourhoods of the unit element. This formulation shows that we have a subgroup topology in the sense of Marshall Hall [2]. We note also that the closure of F (u) is the whole group F, since for any element g, and any n, g< n) belongs to F^\ THEOREM 2. Every endomorphism of F is continuous. Let e be an endomorphism of F. It is sufficient to prove that e is continuous at the unit element; which is to say, that given N we can find n such that ^n) = 1 implies (c(go) = 1. But the mapping <7->(e(gr)) S) is a homomorphism of F into the free group F {N \ so that this is a case of Theorem 1. COROLLARY 1. The closure of a characteristic, or fully invariant, subgroup of F is characteristic, or fully invariant. COROLLARY 2. An endomorphism e of F is determined by the elements For the effect of c on x l} x 2,... determines its effect on any element of F^\ the subgroup they generate, and therefore by continuity on any element of F, the closure of F {t>). With regard to Corollary 1 it may be remarked that a characteristic subgroup of F is not necessarily closed. Consider for instance the element z defined by = [x v x 2 ][x 3, s 4 ]... [x 2n _ lt x 2n ], where [x, y] is the commutator of x and y. Obviously z* 2 ' 0 belongs to the derived group of F for all n, and therefore z belongs to the closure of the derived group. But z does not belong to the derived group. For if it did, it would be the product of a certain number r of commutators, and hence z< 2n) also would be a product of r commutators for all n. This is impossible, because z (2n) is not a product of fewer than n commutators. In fact if, following Magnus [7], we represent J^(n) as a group of formal power series in non-commuting variables, by putting x ( = 1+% and xr 1 = 1 ^-fa,- 2..., a product of r commutators is represented by a series 1+Sa li a,a y +..., where the skew-symmetric matrix (a,,) has rank at most 2r. The corresponding matrix for the element z n) has rank 2n. Thus the derived group of F is not closed. This contrasts with the behaviour of the unrestricted

5 UNRESTRICTED FREE PRODUCTS. 77 direct product of a countable infinity of free cyclic groups, in which every characteristic subgroup is closed*. By Corollary 2 above, in order to know all the ondomorphisms of F it is sufficient to know what sequences g v g 2,... are images of the sequence x lf x 2,... under endomorphisms of F. A necessary condition is that g n -> 1 as 7i->oo; and we shall prove next that this is also sufficient. It is convenient to prove a lemma first. LEMMA 3. The necessary and sufficient condition tjiat the, sequence h v h 2,... of elements of F tends to a limit is that JIT 1 h i+1 tends to 1. The necessity is obvious. To prove the sufficiency assume that the condition holds. Then for n fixed, {hr 1 h i+1 Y" ) = 1 for large t, which is to say that A[ ; ) is constant, and equal to Ji {n) say, for large i. The h Ot) are easily seen to be the components of an element h of F, to which the sequence tends. THEOREM 3. If the sequence g x, g 2,... of elements of F tends to 1, tiien there exists an endormorpjiism of F mapping x ( on g {, i 1, 2,... The group F (m) is generated freely by x lt x 2,..., so that the mapping x i~ > 9i can be extended to a homomorphism y of F^ into F. To show that this can be extended by continuity to an endomorphism of F we must show first that if a sequence of elements of F M tends to a limit in F so does the sequence of their images under y, and secondly that if two such sequences tend to the same limit so do their images. To prove the second of these it is plainly sufficient to show that if h r > 1 then y (/&,)-> 1, and by Lemma 3 this is sufficient to prove the first also. Now to say that y (&,)-» 1 is to say that given n we can choose N such that if i > N, (y(fy)) = 1. Given n, we first choose m so that if i ^ m, g[ n) = 1, as is possible since <7,-» 1. Next, h { is a word, wfaj) say, in the letters x i and their inverses, and since A,-> 1, we can choose N so that if i ^ N, w,(a;,) becomes 1 if we make x i = 1 for j^m. But y{h t ) w t (g i ) i and (y(a,.)) (n) = «;,-(gfj n) ); and since J n) =l if () J j > m, (y(a,)) n = 1 if * > N, as required. Thus h r > 1 implies y{h { )-+1, and the theorem follows. 5. Varieties of topological groups. Denote by X the set of elements x v x 2,... and the unit element with their topology. The group F and the space X have the following properties, as we have proved: (i) X is a subspace of F containing the unit element; (ii) the closure of the subgroup generated by X is F; (iii) any continuous mapping of X into F which maps the unit element on itself can be extended to a continuous endomorphism of F. * This is an immediate cousetjucme of Speckcr [10], lemma in the proof of Satz L

6 78 G. &OMAN To put these facts in perspective, we recall some others. First, F is a free group if there is a set X which generates F and which is such that any mapping of X into any group can be extended to a homomorphism of F into the group. Secondly, a group F which has subset X which generates it, and is such that every mapping of X into F can be extended to an endomorphism of F, determines a variety of algebras, in the sense of P. Hall, which is a sub variety of the variety of all groups. A group G belongs to the variety if every mapping of X into 0 can be extended to a homomorphism of F into 0; and F is a free algebra of the variety. To distinguish them from free groups proper, such groups may be called varietal free groups. Thirdly, F is a free topological group, in the sense of Graev*, if it has a subspace X containing the unit element, such that the closure of the subgroup generated by X is F, and such that every continuous mapping of X into a topological group which maps the unit element on the unit element can be extended to a homomorphism of F into the group. All this suggests the following definitions. A topological group F is a varietal free topological group if it has a subspace X with properties (i) to (iii) above. The variety of topological groups determined by F is the set of topological groups G such that every continuous mapping of X into G which maps the unit element on the unit element can be extended to a continuous homomorphism of F into G. Thus F, in the special sense of 3-4, is a varietal free topological group. But so also is F (<o) taken in the topology it enjoys as a subset of F. For if the elements g v g 2,... of Theorem 3 belong to F M, so do the elements y(g) for g in F^K Thus y is an endomorphism of F^\ and it is continuous because it can be extended to a continuous endomorphism of F. A similar argument shows that if O is a fully invariant subgroup of JH"), then the union of F M and the closure of G is a varietal free topological group for the same subspace X. We consider now the varieties of topological groups determined by F and Fi u). For convenience we restrict ourselves to topologioal groups in which the neighbourhoods of the unit element have a countable basis. We recall that such a group is called complete if every sequence g v g t,..., such that g^gj-* 1 as t and j tend independently to infinity, tends to a limit. THEOBEM 4. If the group G has a countable base of neighbourhoods of the unit element, each of which is a normal subgroup of G, then G belongs to the variety determined by F^K If G is also complete, it belongs to the variety determined by F. A continuous mapping of X into G in which the unit elements correspond can certainly be extended to a homomorphism of F M into G, and to prove * [1]. This differs from the earlier sense of Markoff [8], in which the unit element is supposed not to belong to the closure of X.

7 UNRESTRICTED FREE PRODUCTS. 79 the first part of the theorem we have only to show that this mapping is continuous. As usual, it suffices to prove continuity at the unit element. The proof parallels exactly the second part of the proof of Theorem 3, and we do not repeat it. If 0 is complete, as well as having a normal subgroup topology, it is easy to prove the analogue of Lemma 3 for G, and it then follows as in the proof of Theorem 3 that the continuous mapping of F ia>) into G extends to a continuous homomorphism of F into G. I suspect that the converse of Theorem 4 is true, but can only prove a weaker result. THEOREM 5. If G has a countable base of neighbourhoods of the unit element, and belongs to the variety determined by F M, its topology is a subgroup topology. If it belongs to the variety determined by F, it is also complete. Let K x, K 2,... be a base of neighbourhoods of the unit element of G. If, given n, we can find N such that the subgroup {K N } generated by K N is a subset of K n, then the subgroups {K n }, n= 1, 2,..., will serve as a base of neighbourhoods of the unit element, and G has a subgroup topology. If not there exists k { in {K ( }, i = 1, 2,..., such that k { does not tend to 1. Then k t is a product k {1 k i2... k ir, where k u or its inverse belongs to K (. We arrange the elements k (i in a single sequence g lt g 2,..., first according to increasing i, and then according to increasing j. Then if we assume, as we may, that for i <j, Zp^, this sequence tends to 1. But if we set g { = y (a;,) and extend y to a homomorphism of F M into G, y is not continuous. For k( = g m g m+1...g n for some m, n, and m tends to infinity with i. Thus if y { = x m x m+1...x n then y { tends to 1, but y(y { ) = k ( does not. It follows that G does not belong to the variety determined by F^\ This proves the first part of the theorem. Next suppose that G belongs to the variety determined by F. Then it belongs a fortiori to the variety determined by F M y and so has a subgroup topology. Also, if g v g 2,... are such that g^g^x tends to 1, the mapping x x = 1, x i = gi'2 1 g i, i > 1, can bo* extended to a continuous homomorphism of F into G. If we put y { = x t x 2... x ( then y { tends to a limit. Hence so must its image, which is g (. Thus 0 is complete. It follows, for instance, from Theorems 4 and 5 that the additive group of rationals belongs to the variety determined by F 1^ if it is taken in a js-adic topology, but not if it is taken in the real topology. 6. A subgroup off. If n is a fixed integer, we denote by G ln) the closure of the subgroup of F generated by x n+1, x n+2,... It is evident that the subgroup generated by F< n) and G {n) is their free product. This free produot is not the whole of F. For instance the elements i/ a > = [x v sjfo, x 9 \... [x v x n ]

8 80 G. HlOMAN are the components of an element y of F which does not belong to the subgroup generated by F a) and O (1). We shall denote by P the set of elements of F which belong to the subgroup generated by F {a) and (? (/l) for all n. P is a subgroup of F which contains JPH THEOREM 6. A freely irreducible subgroup of P is either a free cyclic group or contains only the unit element. We remind ourselves that a group is freely irreducible if it cannot be written as a free product of proper subgroups. If a group is a free product of two subgroups then it follows from Kurosch's theorem on the subgroups of free products that a freely irreducible subgroup of it which is not a free cyclic group must be conjugate to a subgroup of one of the factors. If H o>) is the intersection of P and G {n), it is easy to see that P is the free product of F (n) and H {u). Since F< n) is a free group, a freely irreducible subgroup of P which is not free cyclic must be conjugate to a subgroup of H {n \ In particular, an element g of such a group must have ^n) = 1. Since this is true for all n, the group contains only the unit element. The interest of Theorem 6 lies in the fact that P is not a free group. It is possible to prove this directly, using the fact that if the elements g v g 2,... of Lemma 1 belong to P so do the elements h { there constructed. We shall however show that P contains countable subgroups which are not free. Consider, for instance, the subgroup 7 generated by x v x Z)... and 2/o> Vv y& > where the y { are constructed as in Lemma 1 to satisfy By the construction, yf = 1, so that y^)_ x = xf. From this it follows that x x, x 2,..., «,_!, y { _ x generate freely a free group 7,-. The groups Y lf Y 2,... are a properly increasing sequence of free groups on a finite number of generators, whose union is Y. Moreover Y ( _ x is not contained in a proper free factor* of 7,, since the factor group of 7 t - by the self-conjugate closure of 7,_ 1 is generated by Xi_ v y t _ x subject to the relation y f _ 2 = af_iyf_j = 1, and so has no free group on one or more generators as free factor. It follows from Theorem 1 of [3] that Y is not free. This example can obviously be generalized. We may take the elements y t to satisfy provided that first the elements x\, x 2,..., x,_i, y,-_x generate freely a subgroup 7,-, and secondly 7,- is not contained in a proper free factor of For the first, it is sufficient that w ( is of non-zero weight in #,-. For the second it is sufficiontf, and necessary, that w ( is not a power of a primitive clement of the free group generated by x ( and y t. * Cj. [3], Lemma 7. f This follows from the theorem of Neumann [9], and arguments based on Magnus [6].

9 UNRESTRICTED FREE PRODUCTS. 81 Finally, we note that although the first example, due to Kurosch [5], of a group which is not free, but has only free cyclic groups for freely irreducible subgroups, is of a similar type to those considered above, it is not a subgroup of P or even of F. It is, in fact, generated by elements x {, y ( subject to the relations y { _ x = [x (, y { ]. Obviously y ( belongs to every group of the lower central series of this group. But if the element g of F belongs to every group of the lower central series of F, then tf n) belongs to every group of the lower central series of F (n \ and so* tf n) = 1. Thus g = 1. References. 1. M. I. Graev, " Free topological groups", Izvestia Akad. Nauk SSSR (ser. mat.)* 12 (1948), M. Hall, jr., " A topology for free groups and related groups ", Annals of Math., 52 (1950), Graham Higman, " Almost free groups ", Proc. London Math. Soc. (3), 1 (1951), A. Kurosch, " Die Untergruppen der freien Produkte von beliebigen Gruppen ", Math. Annalen, 109 (1934), , " Zum Zerlegungsproblem der Theorie der freien Produkte", Ree. Math. Moscow, N.S., 2 (1937), W. Magnus, " liber diskontinuierliche Gruppen mit einer definierenden Relation ". Journal filr Math., 163 (1930), , "Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring ", Math. Annalen, 111 (1935), A. Markoff, " On free topological groups ", Izvestia Akad. Nauk SSSR, 9 (1945), B. H. Neumann, " On the number of generators of a free product ", Journal London Math. Soc, 18 (1943), E. Specker, " Additive Gruppen von Folgen ganzer Zahlen ", Portugaliae Mathematica, 9 (1950), A. Weil, " L'integration dans les groupes topologiques et ses applications", Actualitd* scientifiques et industrieues, 869 (1938). The University, Manchester, 13. ON INFINITE SOLUBLE GROUPS (IV) K. A. HrasoHf. The object of this note % is to record two further (unrelated) properties of infinite soluble groups with maximal condition for subgroups or, as we have called them, fl-groups. The first of these is concerned with elements of finite order, the second with subgroups of finite index, in an &-group. 1. Let 0 D@ 1 D...D / = 1 be a normal series for in which the factor-groups t/ {+1 are cyclic and, if finite, of prime order. We call such a series a weak composition series of ; Magnus [7]. f Received 19 April, 1951; read 19 April, J which is a continuation of the papers [l]-[3] of the same title. JOUB O