Minimax Groups Research Articles

Fitting subgroups of profinite completions of soluble minimax groups

How far does the class of all finite images of a group determine this group? It is known that finitely generated abelian groups are determined by their finite images up to isomorphism whereas finitely generated nilpotent groups in general are not [4]. At least there are only finitely many nonisomorphic such groups with the same finite images [S]. The same is true for polycyclic-by-finite groups [3] and for finite extensions of torsion-free nilpotent minimax groups which are radicable by its spectrum [7]. It is not known if there is a common generalization to soluble minimax groups the Fitting subgroups of which are radicable by their spectrum. In the way Pickel’s theorem [9] was a first step to the proof of the result on polycyclic groups the Main Theorem of this paper might be helpful on the way of a proof of such a more general result.

Soluble groups which are products of nilpotent minimax groups

Soluble groups which are products of nilpotent minimax groups

A Boundary Problem for Group Testing

A previous result [F. K. Hwang, Tamkang J. Math., 2 (1971), pp. 39–44] showed that a minimax group testing algorithm to find d defectives in n items is to test each item individually for $d\geqq 0.5n$. In this paper we improve this result by proving that individual testing is minimax for $d\geqq 0.4n$. We also conjecture that the same is true for $d\geqq \frac{1}{3}n$. On the other hand, we prove that individual testing is not minimax for $d < \frac{1}{3}n$.

On Minimax Groups which are Embeddable in Constructible Groups

On Minimax Groups which are Embeddable in Constructible Groups

Subnormality and quasinormality in soluble minimax groups

Subnormality and quasinormality in soluble minimax groups

Residually Nilpotent Groups

I am grateful to Professor D. J. S. Robinson for pointing out that Lemma 2.1 and Theorem 2.3 are false. A counterexample is provided by Pontryagin's example of an aperiodic indecomposable abelian group of rank 2 with every non-zero element of type (0, 0, 0, …), which can be found in L. Fuchs's Abelian groups (Oxford 1960), pp. 151–152. There is also an error in the proof of Lemma 2.2, which is nevertheless correct, and the statements of Theorems 3.1 and 3.3 and Corollaries 2.5, 3.2 and 3.4 should be amended by the addition of the following: unless r = 1, when G is abelian. All the subsequent results of the paper depend on corollaries to Theorem 2.3 and alternative proofs are outlined below. Theorems 3.1 and 3.3 and their corollaries can be proved by adding the hypothesis that B is free abelian to Lemma 2.1 and that G is polycyclic in Lemma 2.2, Theorem 2.3 and Corollary 2.4. The given proofs then establish Corollary 2.5, which could alternatively be given a short direct proof. A modified form of Theorem 3.8 can be proved from the following result, which is used in place of Corollary 2.4. LEMMA 2.6. Let G be a completely infinite minimax group which is of abelian section rank r. Let k be the greatest integer less than or equal to r(l +log2 r). Then G is completely infinite of length at most k. Proof. Because G is aperiodic, by [7; vol. 2, 10.32, p. 169] there exists a nilpotent normal subgroup F of G such that G/F is abelian-by-finite. By Corollary 2.5, F is nilpotent of class at most r and by [1; 3.6, p. 606], F ɛ d where d is the derived length of F and hence d r log2 r. Because G is of abelian section rank r the aperiodic rank of F is at most r log2 r and so the aperiodic rank of G is at most r(1 + log2 r). Consequently, G ɛ k. Theorem 3.8 and Corollary 3.9 can now be corrected by replacing “at most r” by the following: at most k, where k is the greatest integer less than or equal to r(1 + log2 r). I wish to thank Dr. R. B. J. T. Allenby for his help with these corrections. Note added in proof. See also the review by D. J. S. Robinson, Math. Reviews, 51 (1976), *13042.

Groups whose finite quotients are supersoluble

I f G is an infinite group each of whose finite quotients has some given property, one may ask how this situation is reflected in the global structure of the group G. The first result in this area of investigation was obtained by Hirsch in [5] where he proved that a polycyclic group G each of whose finite quotients is nilpotent must itself be nilpotent. This result has been extended in two directions by D. J. S. Robinson, who showed that the same holds if G is merely assumed to be either a torsion-free by finite soluble minimax group or a finitely generated hyper-(abelian or finite) group ([7] and [S]). On the other hand W’ehrfritz and Platonov proved (independently) that the same result holds under the assumption that G is a linear group over a finitely generated integral domain (see [ 121, 4.16 and 10.15). Now supersolubility is in some ways a subtler property than nilpotency, and knowledge of the groups in the title seems to be rather more fragmentary. Baer [l] has shown that if a group G each of whose finite quotients is supersoluble is polvcvclic, then G must itself be supersoluble; and Wehrfritz [I l] . I has shown that if G is a linear group over a finitely generated integral domain of nonzero characteristic then G is hypercyclic (he has recently generalized this in [ 131; we quote his theorem in full in Section 6). In this paper we extend the former result to larger classes of (roughly) soluble groups, and we settle the characteristic-zero case of the linear group problem.

Subnomality in soluble minimax groups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

The subnormal coalescence of some classes of groups of finite rank

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if wheneverHandKare subnormal -subgroups of a groupGthen their join &lt;H, K&gt; is also a subnormal -subgroup ofG. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

A theorem on finitely generated hyperabelian groups

Our main object is to prove that a finitely generated non-nilpotent hyperabelian group has a finite non-nilpotent homomorphic image. The terminology "here is as follows: if 9 ~ is a group theoretical property, a group G is a hyper-~ group if it has an ascending series (of arbitrary ordinal type beginning at 1 and ending at G) of normal subgroups such that each factor of the series has the property ~: when ~ is inherited by homomorphic images, it is not difficult to see that hyper-~ is equivalent to the requirement that every non-trivial homomorphic image contain a non-trivial normal subgroup with the property ~. Actually we will prove a slight generalization of the result mentioned above. Theorem 1. Let G be a finitely generated hyper-(abelian or finite) group. If G is not nilpotent, it has a finite non-nilpotent homomorphic image. The special case where G is polycyclic is due to Hirsch ([6], Theorem 3.24). At least two generalizations of Hirsch's theorem already exist. A non-nilpotent soluble minimax group without groups of type pO~ in the centre of its Fitting subgroup for any prime p has a finite non-nilpotent homomorphic image (Robinson [7], Theorem A and Theorem 5.11). (A minimax group is a group which has a series of finite length whose factors satisfy either the maximal or minimal condition on subgroups.) A similar result is true for non-nilpotent linear groups over a finitely generated integral domain (cf. Wehrfritz [10], Lemma 2).