If X is a class of groups, the class of counter-Xgroups is defined to consist of all groups having no non-trivial X-quotients. The counter-abelian groups are the perfect groups and the counter-counter-abelian groups are the imperfect groups studied by Berrick and Robinson [2]. This paper is concerned with the class of counter-counterfinite groups. It turns out that these are the groups in which any non-trivial quotient has a non-trivial representation over any finitely generated domain (Theorem 1.1), so we shall call these groups highly representable or HR-groups.

Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).

Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequality

Here

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.

Throughout, R denotes a commutative domain with 1, and Q (≠R) its field of quotients, which is viewed here as an R-module. The symbol K will stand for the R-module Q/R, while R denotes the multiplicative monoid R/0.

Let be either a C*-algebra (with norm ∥ ∥) or a symmetric ideal of operators on a Hilbert space (with norm denoted by σ). Let a1…, an be self-adjoint elements, and let a0 = .

Let G be a finite group, H a copy of its p-Sylow subgroup, and N the normalizer of H in G. A theorem by Nishida [10] states the p-homotopy equivalence of suitable suspensions of BN and BG when H is abelian. Recently, in [3] the authors proved a stronger result: let ΩkH be the subgroup of H generated by elements of order pk or less; if

then BN and BG are stably p-homotopy equivalent. The hypothesis above is obviously verified when H is abelian. In the same paper the authors recall that H does not verify such condition when p = 2 and G = SL2(Fq) for a suitable odd prime power q; in this case BG and BN are not stably 2-homotopy equivalent.

Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

Let R be a regular semigroup and denote by (R) its congruence lattice. For , the kernel of pis the set ker . The relation K on (R) defined by λKp if ker λ = ker p is the kernel relation on (R). In general, K is a complete ∩-congruence but it is not a v-congruence. In view of the importance of the kernel-trace approach to the study of congruences on a regular semigroup (the trace of p is its restriction to idempotents of R), it is of considerable interest to determine necessary and sufficient conditions on R in order for K to be a congruence. This being in general a difficult task, one restricts attention to special classes of regular semigroups. For a background on this subject, consult [1].

In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim A, where codepth A is the length of a maximalA-cosequence and dimA is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeriobtained several properties of co-Cohen-Macaulay Artinian R-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).