In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].

The terminology of [1] will be assumed in what follows. Let Pb(G) stand for the set of pentagons in the graph G. Call a graph pentagongenerated when it is the union of its contained pentagons. Let P5,3 be the class of connected trivalent pentagon-generated graphs with girth 5. These graphs form a family including the Petersen graph and the graph of the dodecahedron. They are studied here and completely classified in terms of a decomposition which all but some specifically determined indecomposable graphs admit.

Assume henceforth that H ∈ P5,3. Let Ek(H) be the set of edges in exactly k ∈ 0 pentagons of H. Clearly Ek(H) = 0 if k ≠ 1, 2, 3, 4 and |E1(H) ∩ E(P)r ≦ 2, for all P ∈ P5(H). P ∈ P5(H) is singular when |E1(H) ∩ E(P)r = 2,.

Let p be a prime and P be a p-subgroup of a finite group G. Suppose that g ∈ G and that P ∩ Pg has index p in P. In [4], we assumed that g normalizes no non-identity normal subgroup of P. We obtained some bounds on the order of P and some applications to the case in which p = 2 and P is a Sylow 2-subgroup of 〈P, Pg〉. In this paper, we examine this situation further by considering the isomorphism ϕ of P ∩ Pg-l onto P ∩ Pg given by ϕ(x) = xg. We actually consider arbitrary isomorphisms ϕ between two subgroups of index p in P. However, an easy argument (Lemma 2.3) shows that every such ϕ can be obtained as above for some G and some g. We obtain some results concerning the nilpotence class rather than the order of P.

In this paper we use the Bourbaki [2] conventions for rings and modules. All rings are associative but not necessarily commutative and have a 1; all modules are unital.

Bass [1] calls a ring A left perfect if and only if every left A -module has a projective cover, which he shows is equivalent to every flat left A -module being projective. Bass calls a ring A semi-perfect if and only if every finitely generated module has a projective cover and shows that this concept is leftright symmetric.

We will define a ring A to be quasi-perfect if and only if every finitely generated flat left A -module is projective.

An exercise [6, Exercise 10, p. 136] is given by Lambek to show that every semi-perfect ring is quasi-perfect.

A topological group G is called monothetic if it contains a dense cyclic subgroup. An element x of G is called a generator of G if x generates a dense cyclic subgroup of G. We denote by E(G) the set of generators of G; the complement of E(G) in G, consisting of the “non-generators” of G, we write as N(G) Throughout this paper we consider only locally compact abelian (LCA) groups satisfying the T2 separation axiom (note that a monothetic group is automatically abelian). In [1] certain problems of measurability concerning the set E(G) are discussed. In this paper we shall consider some algebraic and topological properties of the sets E(G) and N(G)

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.

If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

Let M be the space of all bounded regular complex-valued Borel measures defined on I = [0, ∞). M is a Banach space with ‖μ‖ = ∫d|μ|(x) (μ ∈ M). (Integrals in this paper extend over all of I unless otherwise specified.) Let v be a fixed real number no smaller than and let if z ≠ 0 and , where Jv, is the Bessel function of the first kind of order v and cv =[2vΓ(v + 1)]–1; is an entire function, as can be seen from the power series definition of

The Hankel-Stieltjes transform of order v is given by .

Suppose that we are given an isomorphism ϕ between two subgroups of index p in a finite p-group P. Let N(ϕ) be the largest subgroup of P fixed by ϕ. By a result of Sims [2, Proposition 2.1], n(ϕ) is a normal subgroup of P. In [2], we showed that P/N(ϕ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?

In this paper, we consider characteristic subgroups of P rather than normal subgroups.

In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

A biorthogonal sequence is a double sequence (xi,fi) where each xi is from some locally convex space X, each fi is from X* and fi(xj) = δij. A biorthogonal sequence is called total if the functionals (fi) are total over X and is called fundamental if sp(xi) is dense in X. If a biorthogonal sequence is both total and fundamental we refer to it as a Markushivich basis or, more simply, an M-basis.

If (xi,fi) is a total biorthogonal sequence for X, then X can be identified with the space of all scalar sequences (fi(x)) under the correspondence x ↔ (fi(x)). We refer to this space as the associated sequence space with respect to (xi, fi). With this correspondence, xi corresponds to and fi corresponds to Ei, the ith coordinate functional.

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type

1.1

where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).

Let S be a topological semigroup (i.e., S is a semigroup with a Hausdorff topology such that the mapping from S × S to S defined by (s, t) → s ⋅ t for all s, t in S is continuous when S × S has the product topology) and C(S) be the space of bounded continuous real valued functions on S. For each ƒ in C(S) and a in S, define || ƒ || = sup {|ƒ(s)|: s ∈ S} (sup norm of ƒ); raƒ(s) = ƒ(sa) and laƒ(s) = ƒ(as) for all s in S.

In [6, p. 586] Spector asked whether given a number e there exists a unary partial function from the natural numbers into {0, 1} with coinfinite domain such that for any function ƒ into {0, 1} extending it is the case that

We answer this question affirmatively in Corollary 1 below and show that can be made partial recursive (p.r.) with recursive domain. The reader who is familiar with Spector's paper [6] will find the new trick that is required in the first paragraph of the proof of Lemma 2 below.

From one point of view, this is a theorem about trees which branch twice at every node. We shall formulate a generalization which applies to trees which branch n times at every node.

A space X is in-expandable, where m is an infinite cardinal, if for every locally finite collection {Hα| α ∈ A} of subsets of X with |A| ≦ m(cardinality of A ≦ m) there exists a locally finite collection of open subsets {Gα| α ∈ A} such that Hα ⊆ Gα for every α ∈ A. X is expandable if it is m-expandable for every cardinal m. The notion of expandability is closely related to that of collection wise normality introduced by Bing [1], X is collectionwise normal if for every discrete collection of subsets {Hα|α ∈ A} there is a discrete collection of open subsets {Gα|α ∈ A} such that Hα ⊆ Gα for every α ∈ A. Expandable spaces share many of the properties possessed by collectionwise normal spaces. For example, an expandable developable space is metrizable and an expandable metacompact space is paracompact.

The purpose of this paper is to extend the theorems in [3; 7] to uniform spaces and to prove some additional theorems. These results are related to [4; 5]. Notation and definitions are as in the book [2]. For a general reference on nets see [6]. All topological spaces are assumed to be Hausdorff.

THEOREM 1. Let (X, T, Π) be a transformation group, where X is a locally compact, locally connected, uniform space. Let E denote the set of all points at which T is equicontinuous and N = X – E. Let N be closed totally disconnected and each orbit closure in N be compact and let E be connected. Then N contains at most two minimal sets. (Note: We will assume that N ≠ ∅ so that N will contain at least one minimal set.)

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.

Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction (UCED), was first used by A. L. Garkavi [5; 6] to characterize normed linear spaces for which every bounded subset has at most one Čebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].

In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form

*

where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.

Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],