Group GI Research Articles

Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups

1.1 Let G be a profinite group and p be a fixed prime. In this paper we will be concerned with H∗ c (G;Fp), the continuous cohomology of G with coefficients in the trivial module Fp. We will abbreviate H∗ c (G;Fp) by H∗(G;Fp), or simply by H∗G if p is understood from the context. We recall that if G is the (inverse) limit of finite groups Gi then H∗G = colim H∗Gi. Throughout this paper we will assume that H∗G is finitely generated as Fp algebra. By work of Lazard [La] it is known that this holds for many interesting groups, for example for profinite p analytic groups like GL(n,Zp), the general linear groups over the p adic integers. In case H∗G is finitely generated as Fp algebra Quillen has shown [Q1] that there are only finitely many conjugacy classes of elementary abelian p subgroups of G (i.e. groups isomorphic to (Z/p) for some natural number n). In other words, the following category A(G) is equivalent to a finite category: objects of A(G) are all elementary abelian p subgroups of G, and if E1 and E2 are elementary abelian p subgroups of G, then the set of morphisms from E1 to E2 in A(G) consists precisely of those homomorphisms α : E1 −→ E2 of abelian groups for which there exists an element g ∈ G with α(e) = geg−1 ∀e ∈ E1. The category A(G) plays an important role both in Quillen’s results and in the work presented here. This category entered into Quillen’s work as follows. The assignment E 7→ H∗E extends to a functor from the opposite category A(G)op to graded Fp algebras and the restriction homomorphisms H∗G −→ H∗E (for E running through the elementary abelian p subgroups of G) induce a canonical map of algebras q : H∗G −→ limA(G)op H∗E.

The isomorphism problem for a class of para-free groups

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

Strong bands of groups of left quotients

An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

Right Chain Domains with Prescribed Value Holoid

Let H = Hi ∈ I (Gi, Ni) be the split holoid of a family of ordered groups Gi with convex subsets Ni that contain G+i, the positive cone of Gi, where I is an ordered index set. Then there exists a right invariant right chain domain R with H ≅ H(R), the associated value semigroup of nonzero principal right ideals of R.

On soluble groups with finite homomorphic images of bounded rank

Let G be a group. We denote by 9r(G) the set of all finite homomorphic images of G. The study of the connection between the properties of G and of groups of the set ~'(G) is a traditional problem of group theory. In this connection the following question, proposed by Wilson (see [1, Question 12.90]), is of interest: is a finitely generated residually finite group G minimax if there exists a finitely generated soluble minimax group H such that 9V(G) = 9V(H)? In the present paper we show that G is also minimax if ~-(G) C_ ~'(H). We recall that a group is minimax if it has a finite subnormal series each factor of which satisfies either the minimality condition or the maximality condition for subgroups. The basic properties of minimax groups are discussed in [2]. Lemma 1. Let G be a residually finite group, and let H be a group of finite rank. If ~(G) C ~(H), then G does not contain infinite elementary Abelian p-subgroups. Proof. Suppose that G contains an infinite elementary Abelian p-subgroup A. Then G contains a finite elementary Abelian p-subgroup B such that [B I > pr, where r is the rank of H. Since G is a residually finite group, we see that G contains a normal subgroup R of finite index such that B fq R = 1. Then G = G/R contains an elementary Abelian p-subgroup/~ such that IBI > p~. Since G E 37(G) and ~-(G) C ~-(g), we have G E 5r(g). Therefore, r(G) pL The lemma has been proved. Lemma 2. Let G be a subgroup of the Cartesian product G = :~iGi of finite soluble groups Gi of bounded orders. Suppose that H is a group of finite rank. If ~(G) C 3U(H), then G is a finite group. Proof. Since the orders of Gi are bounded, it follows that the solubility lengths of Gi are also bounded and, hence, G is soluble. Suppose that G is an infinite group and show by induction on the solubility length of G that the latter contains an infinite elementary Abelian p-subgroup. Assume that G is an Abelian group. Since there exists n E N such that IGil < n for all i, it follows that there exists t E N such that G~ = 1 for all i; therefore, G t = 1. From this it follows that G contains an infinite elementary Abelian p-subgroup. Let D be a derived subgroup of G. If ]D I = :xD, then by the inductive hypothesis G contains an infinite elementary Abelian p-subgroup. Thus, one can assume that IDI < :xD. Since G is residually finite, there exists a normal subgroup R < G of finite index such that RND = 1 and, consequently, R is an infinite Abelian subgroup of G. As was shown above, R contains an infinite elementary Abelian p-subgroup. Thus, G contains an infinite elementary Abelian p-subgroup. This contradicts Lemma 1. Therefore, the group G is finite. The lemma has been proved. Lemma 3. Let G be a residually finite group, and let H be an almost metanilpotent soluble group of finite rank. If ~(G) C F(H), then G is an almost metanilpotent group. Proof. Since G is a residually finite group, we see that G is a subgroup of the Cartesian product = ~iGi, where Gi E JZ(G). The group H has the normal series

Minimal relative relation modules of finite 𝑝-groups

Consider 1 → S → E → G → 1 1 \to S \to E \to G \to 1 , where G G is a finite p p -group generated by g i , 1 ⩽ i ⩽ d {g_i},\;1 \leqslant i \leqslant d , and E E a free product of cyclic groups ⟨ g i ⟩ , 1 ⩽ i ⩽ d \langle {g_i}\rangle ,1 \leqslant i \leqslant d . If d d is the minimum number of generators for G G , then we prove that the largest elementary abelian p p -quotient S / S ′ S p S/{S’}{S^p} , regarded as an F p G {\mathbb {F}_p}G -module via conjugation in E E , is nonprojective and indecomposable.

Combined effects of nicardipine and hypocapnic alkalosis on cerebral vasomotor activity and intracranial pressure in man

The effect of hypocapnic alkalosis (HA) on nicardipine-induced cerebral vasodilatation was studied in 2 groups of patients undergoing stereotaxic brain biopsy under general anaesthesia. Arterial diameter (AD) was measured in 16 different locations on a carotid arteriogram (lateral view), and intracranial pressure (ICP) was recorded with an intraventricular catheter. At time T0, in normocapnia an arteriogram was performed in both groups. The first group (GI) was then studied in hypocapnia (T1) and following an injection of nicardipine (T2), while the second group (GII) was studied first after injection of nicardipine (T1) and then in hypocapnia (T2). Groups GI (n = 6; 44 y) and GII (n = 6; 46 y) were similar with regard to age, blood pressure, heart rate and PaO2 at all three phases of the study. HA caused a 9.5% decrease in AD (GI.T1) compared to baseline values, and a 15.2% decrease when preceded by injection of nicardipine (GII.T2). In the latter case the decrease was 3% in comparison with baseline. Nicardipine increased AD by 14.7% (GII.T1) and by 18.5% when preceded by HA (GI.T2), but the rise (7.3%) was not significant in comparison with the baselines value. The changes variations were similar whether the entire arterial trunk or only the supraclinoid region were studied. HA decreased ICP by 44% (GI.T1) and by 50% after nicardipine (GII.T2). Nicardipine did not cause an increase in ICP. Nicardipine and HA antagonise each others vasomotor effects, as previously shown in the baboon using nimodipine.

Covering étendues and Freyd’s theorem

From Freyd's covering theorem for Grothendieck topoi, it immediately follows that every Grothendieck topos E admits a hyperconnected geometric morphism 3 -C, where 7 is an etendue of (discrete) G-sheaves. As a corollary, we obtain that E admits an open surjection from a localic topos. A fundamental representation theorem in topos theory is provided by the following result of P. Freyd. If E is a Grothendieck topos, there is a connected atomic geometric morphism 7 -f &, where 7 is localic over a topos C(G) of continuous G-sets, G being a topological group. In fact, G may be chosen to be either Go, the group of permutations of N, or G1, the group of order-preserving permutations of Q. (See [1] or the excellent survey article [2].) In [4], it was shown that given 6, there is a hyperconnected geometric morphism 7 6, where 7 is an etendue, i.e., there is a surjective local homeomorphism F/X with F/X localic. Using Freyd's result, we not only obtain this theorem directly (with . a particularly nice etendue), but the fact that & has a localic cover via an open surjection [3, p. 61] also follows. To fix some notation, sh(G; B) denotes the topos of (discrete) G-sheaves, where B is a locale on which the group G acts. sh(G; B) is a basic example of etendue and is locally sh(B). Also, SG denotes the topos of G-sets. There is a canonical map SG -A C(G) [4]. THEOREM. Let E be a Grothendieck topos. There is a hyperconnected geometric h morphism 7 -h 6, where 1 is an e'tendue of G-sheaves, sh(G; B). (G can be taken to be either the group Go or Gi.) PROOF. Let C(G)[A] I+ & be connected and atomic, where C(G)[A] is the topos of C(G)-sheaves on an internal locale A in C(G). Let B be the Macneille completion of A. Thus A is the cocontinuous reflection of B along the hyperconnected geometric morphism SG -9 C(G). The following diagram can be seen to be a pullback. sh(G; B) 9 ) (G) [A] 1 l~~~~k Sc G C(G) Since pullbacks perserve hyperconnectedness [3, p. 54], 9 is hyperconnected. Connected and atomic implies hyperconnected [2, Lemma 2.1], thus the composite f sh(G; B) J9 C(G)[A] f E is hyperconnected. Take this to be h. O Received by the editors January 31, 1986. 1980 Mathematics S*ject ai&fication (1985 Revison). Primary 18B25.

Self-duality and torsion Galois modules in number fields

Let K/k be a Galois extension of number fields with Galois group r, and let D, o be the rings of integers of K, k, respectively. In this paper we investigate the structure of 0 as an or-module by comparing 0 with its dual, Horn@, 0). M. J. Taylor [ 11, Theorem, p. 1731 proved in 1978 that these two modules are stably isomorphic over Zr, assuming that all the primes of o dividing the order of r are unramified in K. This result had been conjectured by Frohlich [5, p. 4231 in the case k = Q. Its generalization to all tame extensions is an easy consequence of Taylor’s subsequent proof of Frohlich’s conjecture describing the class of 0 in the class group of locally free ZTmodules in terms of the Artin root number [ 12, Theorem 1, p. 4 1; 5, Proposition 3, p. 4391. To study the Galois module structure of 0 in the wild case Queyrut [7] introduced the S-Grothendieck groups Gi(oT) and K”(oT), where S is a set of primes of o. Namely, G&(oT) is the Grothendieck group corresponding to the category of finitely generated o-torsion free or-modules, with relations arising from short exact sequences splitting outside S; Ki(oT) is the Grothendieck group of all finitely generated o-torsion free or-modules which are locally projective outside S, with relations arising from short exact sequences. With Cassou-Nogues [3, Corollaire 6.3, p. 231, Queyrut generalized Taylor’s duality result to the wild case by showing that if S is a set of rational primes containing those with a divisor in o wildly ramified in K, then the class of 0 is equal to the class of its dual in K:(U). As in the tame case, this result could be obtained as a consequence of Queyrut’s conjecture concerning the class of 0 in K~(ZI’) [3, p. 81, which was proved in several cases [3]. The proofs of the above results used the description of certain Grothendieck groups as quotients of groups of character functions, as well as results on modules introduced by Swan in [lo].

Resistance to Extinction After Gradually Increased or Decreased Ratio of Partial Reward Schedule

Resistance to extinction of the gradually increased (gI) or decreased (gD) ratio of partial reinforcement has been investigated in order to assess effectively partial-reinforcement theories. COGAN et al. (1975) reported greater resistance in gD than in gI employing an extensive acquisition training and the sucrose solution as a reinforcer, and interpreted the results as supporting AMSEL'S frustration hypothesis. ISHIDA (1978), however, found the opposite result with food reward after a moderate number of acquisition trials. Thus, the present study was desingned to determine the resistance to extinction after gI and gD schedules under appropriately arranged experimental conditions, and to assess partial-reinfocement hypotheses.Three groups of 9 rats each were trained with extensive acquisition trials (12 trials a day for 9 days) and then extinguished (12 trials a day for 6 days) in a straight alley. Groups gI and gD received the reward sequences as shown in Table 1. Group RA, which was added as a control, received random sequence. All three groups were rewarded at 53% with food pellets.There was not a significant difference among groups during acquisition, although performance of gI was below those of the other two groups in early trials (Fig. 1). During extinction three groups differed significantly, i. e., group gI was most resistant to extinction, RA was less, and gD was least (Fig. 2).The extinction results did not support the AMSEL'S view that the later occuring nonreinforcements (the group of gD) should produce greater frustration and greatest resistance to extinction. The present findings are interpreted both by reinforcement-level view (CAPALDI, 1978) and attention theory (SUTHERLAND and MACKINTOSH, 1971). However, it is suggested that CAPALDI'S thesis is more appropriate one, considering the predictability of the results, especially the difference between RA and each of gI and gD.

Origin of the Heterogeneity of Peroxidase Isoenzyme Group Gr from Nicotiana tabacum. I. Conformation

Summary By several methods of de- and renaturation (treatment with urea, guanidine and mercaptoethanol, freezing-thawing, heating-cooling etc.) it was studied whether the multiple molecular forms of tobacco peroxidases, group GI, are conformers or not. Most of the applied methods resulted in inactivation of the enzymes only, or did not show any effect. Only heating-cooling resulted in distinct denaturation-renaturation curves but there was no change in isoenzyme pattern at all. Also treatment of GI-isoenzymes with ion-exchange resins or glycosidases did not influence the molecules in any way. Therefore it was concluded that GI-isoenzymes are not conformational variants of a single polypeptide chain and that the origin of heterogeneity might be -due to different primary structures of the isoenzymes.

About sequences of maximal subgroups: a few answers to a question from Bertaut

To each subgroup gi of the space group G does not necessarily correspond a derivative structure si (space group gi) of the structure S (space group G). This paper discusses some symmetrical, geometrical and thermodynamical aspects which occur in the real existence of a derivative structure.

サカナにおける部分強化の配合型と消去抵抗の関係

Goldfish were trained and extinguished in an alley-type apparatus in order to test the assumption that the PRE could be found even in fish by strengthening the rate of conditioning to the stimulus aftereffect of nonreinforcement. There were 3 groups: 2 partial reinforcement groups in which reinforcement ratios were gradually increased (Group gI) or decreased (Group gD), and one continuous reinforcement (control) group. The PRE was not found in Group gI. Group gD showed the least resistance to extinction, These results suggest that resistance to extinction is influenced mainly by a series of reinforced trials very close to the extinction period, but not influenced by the aftereffect of nonreinforcement.

Neue Vorstellungen zum Problem der Isoperoxidasen anhand der Trennung durch Disk-Elektrophorese und isoelektrische Fokussierung

Isoelectric focusing (IEF) of peroxidases of different organs and tissues of Nicotiana tabacum L. was performed on thin-layers of Sephadex and polyacrylamide. Isoelectric points (pI's) of peroxidase bands were measured by special electrodes. — The two types of layers showed very similar results. Reproductibility of pI's was better on polyacrylamide. This method is also easier to practise and requires less time than IEF on Sephadex (3 h versus 18). Thus for analytical purposes the acrylamide-technique is preferable, but if it is necessary to regain the separated enzymes it is better to perform IEF on Sephadex. — When IEF-patterns of peroxidase are compared with the disk electrophoresis (DE) patterns of the same tissue, important differences are observed. The 6 bands of GI (fast migrating, anodic group of DE) are reduced to 2 on the strongly acidic side of IEF (independent of the tissue studied). That means only 2 proteins in GI can be separated by pI's of the molecules. Maybe the heterogeneity of GI bands after DE indicates the presence of conformational isomers (conformers). — Because of the reduced number of bands in GI after IEF there is no difference in the patterns of many tissues (flowers, leaves, shoots, pith) as there is after disk electrophoresis. In the case of GII (slow migrating, anodic group of DE) on the other hand, there are always 4 different bands after isoelectric focusing in the lower acid region instead of 3 after disk electrophoresis. Disk electrophoresis of peroxidase-groups separated by isoelectric focusing shows the same patterns as direct disk electrophoresis of the extract. The methods produce no artifacts. —Comparison of these results with the peroxidase-patterns of tobacco found by other workers and by other techniques leads to the conclusion that there exist at least 4 “isoenzymes” of peroxidase corresponding to the 4 groups GI, GII, GIII, and GIV.

The Nakayama Map and Ramification for Maximally Complete Fields

Let K be a maximally complete valued field and let L be a totally ramified Galois extension of K with Galois group G. Assume (i) the value group quotient of L|K is cyclic and (ii) there exists an unramified cyclic extension of K of the same degree as L. Then there is an isomorphism of Ga onto a subgroup A/N(L×) of K×/N(L×) which maps the ramification group Gi onto AiN(L×)/N(L×) for all i &gt; 0 where Ai = {x ∊ A|v(x ‒ 1) ≧ i}. This generalizes certain results of Local Class Field Theory.

The space of conjugacy classes of a topological group

The space G# of conjugacy classes of a topological group G is the orbit space of the action of G on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic [ZJgroups, the universal covering space of G# may be obtained as the space of conjugacy classes of a group which is locally isomorphic with G, and the Poincare group of G# is found to be isomorphic with that of GIG', the commutator quotient group. In particular, it is shown that the space G# of a compact analytic group G is simply connected if and only if G is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case. I. Defimitions and elementary properties. Two elements x, y of a topological group G are called conjugate, and we write x t y, if there is an element t E G such that y = txt-1. The equivalence class of a point x under this relation is called the conjugacy class of x, denoted Ix. A subset of G which is a union of conjugacy classes is invariant under inner automorphisms and will be said to be invariant. If G acts on itself by inner automorphisms, the inner automorphisms determined by the center Z(G) of G are trivial and G/Z(G) acts effectively on G. The orbit space under the action of G or GIZ(G) is called the space of conjugacy classes of G, denoted G#. If G is the direct product of groups Gi, then G# is homeomorphic with the Cartesian product of the spaces Gf (see [5, p. 130]). The space G# of a compact analytic group G is homeomorphic with the orbit space TIW of the action of the Weyl group W on a maximal toroid T of G [1, p. 95]. If G is semisimple, G# may be obtained by identifying certain boundary points of a compact convex polyhedron in the Lie algebra of T (see [2, Example 6]). Some elementary proofs and [11, p. 231] give the following: LEMmA 1. If G is a compact analytic group, then G# is compact, Hausdorff, second countable, and locally arcwise simply connected. Received by the editors June 1, 1973 and, in revised form, November 5, 1973. AMS (MOS) subject classifications (1970). Primary 57F99; Secondary 54H25, 20F35.

Group Partition, Factorization and the Vector Covering Problem

The covering problem. Let Si(i = 1, 2,…,n) be given sets containing mi elements respectively and let1be their cartesian product. The elements of S(n) will be called vectors. The vector (x1 x2,…, xn) covers (y1 y2,…, yn) if xi =yi for at least n—1 values of i. A subset M of S(n) is said to be a covering (perfect covering) of S(n) if each member of S(n) is covered by at least (exactly) one member of M. A covering M is said to be linear if the sets Si are groups Gi and M is a subgroup of G(n) = S(n) Denote by σ(n; m1 m2,…, mn) the value of min |M| when M runs through all coverings of S(n) and by σ(n; m1 m2,…, mn) the value of min |M| when the sets Si are given groups Gi and M runs through all linear coverings of G(n).

A subgroup theorem for free products of pro-finite groups

Fakzrltcit fiir Mathematik und Injoormatik, Universitiit, 68, Mannheim, West Gwmaq~ Communicated by P. A%f. Cohn Received July 13, 1970 The notion of a free product of pro-finite groups has some important applications in the theory of algebraic number fields (see [3]). In this connection, it is interesting to get some knowledge about the subgroups of such a free product. The aim of this paper is to show a theorem for the open subgroups of a free pro-finite product, which is an analog of Kurosh’s well- known subgroup theorem for the free products of discrete groups in the usual sense. By K we denote a class of finite groups, which is closed under formation of subgroups, factorgroups and group extensions (c.g., all finite groups, the solvable groups, the p-groups, etc.) By a puo-@roup 6 we understand a projective limit 6 7 I;m Gi of groups Gi E (5. Let 6, , E E 81, be a family of pro-K-groups. A family of homomorphisms 7% : Cc,, + -5 into a pro-K-group $j is called convergent, if every open subgroup of $ contains almost every (i.e., up to a finite number) of the images ~~(05~). This condition is clearly empty, if VI is a finite index set. The free pro-K- pvoducc of the pro-K-groups 6, is now defined as to be a pro-K-group (5 =: JJ&,[ . 6, , together with a convergent family of homomorphisms with the following universal property: If y,? : (ci,> ---f & is any convergent 104

Well-known ${\rm LCA}$ groups characterized by their closed subgroups

In this paper we determine (1) the class of all non- discrete LCA groups for which every proper closed subgroup is the kernel of a continuous character of the group, (2) the class of locally compact groups whose closed subgroups are totally ordered by in- clusion, and (3) the class of infinite LCA groups whose proper closed subgroups are topologically isomorphic. Since all these de- terminations involve only the most common LCA groups, we may regard our findings as characterizations of natural classes of these well-known groups. The program of deriving information about locally compact Abe- lian (LCA) groups from a knowledge of their closed subgroups has received attention in recent years; see, for example, references (3), (4), and (5). In this paper we shall state and prove three theorems characterizing some of the most common LCA groups by means of very natural hypotheses upon their closed subgroups. The LCA groups of which we shall make constant use are the circle T, the additive real numbers R, the integers Z, the cyclic groups Z{n), the quasicyclic groups Z(p°°), the £-adic integers JP and the p-adic numbers Fv. Precise definitions of all these groups may be found in (l); in particular, much detailed information on the groups Jp and Fp may be found in (l, §§10 and 25). Except where explicitly stated, all groups throughout are assumed to be LCA and Hausdorff topological groups. If the group Gi is topologically isomorphic to the group G2, we write Gi=G2. The character group of a group G is de- noted by G, and the kernel of a character 7 is written as ker y. Finally, if x is an element of a group G, we use the symbol (x) to denote the subgroup of G generated by x; the closure of this subgroup is written as (x). These findings constitute part of the author's doctoral disserta- tion submitted to Stanford University in 1969. The author is grateful to the National Science Foundation for financial support and to

On the Decomposition of Groups

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.