In this paper we are concerned with the representation theory of the symmetric groupsover a field K of characteristic p. Every field is a splitting field for the symmetric groups. Consequently, in order to study the modular representation theory of these groups, it is sufficient to work over the prime fields. However, we take K to be an arbitrary field of characteristic p, since the presentation of the results is not affected by this choice. Sn denotes the group of permutations of {x1, …, xn], where x1,…,xn are independent indeterminates over K. The group algebra of Sn with coefficients in K is denoted by Фn.

A graph Gn consists of n distinct vertices x1x2, …, xn some pairs of which are joined by an edge. We stipulate that at most one edge joins any two vertices and that no edge joins a vertex to itself. If xi, and xj are joined by an edge, we denote this by writing xi ∘ xj.

Consider a second graph HN, where n ≦ N. Following Rado [1], we say that a one-to-one mapping/of the vertices of Gn into the vertices of HN defines an embedding if xi ∘ xj implies f(xi) ∘ f(xj), and conversely, for all i, j = 1, 2,…, n. If there exists an embedding of Gn into HN, we denote this by writing Gn <HN. The particular graph HN is said to be n-universal if Gn < HN for every graph Gn with n vertices.

The use of graphs in the study of groups is well-established. In this paper, we wish to indicate how certain graph-like objects may be used in a similar way. A diagram is a pseudograph which may have some free edges, i.e. edges with just one end.

In part I of this paper P. Hall's formula for finite stem groups was derived. Using results of C. R. Leedham-Green and S. McKay, a similar formula for isoclinic groups with arbitrary branch factor group is shown.

The main result of this paper is the following theorem, which appears without proof in [1, p. 203].

In recent years there have appeared solutions of several integral equations of the type

in which the kernel K(x) contains (as a factor) one of the classical orthogonal polynomials or a hypergeometric function.

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of

(a) all antichains;

(b) all quasi-complete chains;

(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);

(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;

(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);

(f) the six-element crown C6 with Hasse diagram

A similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

This note establishes two statements from R. M. Fossum's review [1] of a paper by E. A. Magarian [2]. Firstly, if A → B is a pure homomorphism (of commutative rings) then A[[x1,…,xs]] → B[[x1,…, xs]] is pure. Secondly, if Rn → R is a directed family of pure homomorphisms then ∪ Rn → R is pure. A consequence is that if Rn → R is a directed family of pure homomorphisms and if R is Noetherian, then ∪ Rn[[x1,…,xs]]is Noetherian.

A free product sixth-group (FPS-group) is, roughly speaking, a free product of groups with a number of additional defining relators, where, if two of these relators have a subword in common, then the length of this subword is less than one sixth of the lengths of either of the two relators.

Britton [1,2] has proved a general algebraic result for FPS-groups and has used this result in a discussion of the word problem for such groups.

Let S be a closed Riemann surface of genus

g > 1,

so that Ŝ, the universal covering surface of S, is hyperbolic. We can then uniformize S by a discrete, nonabelian group Γ1 of Möbius transformations of the upper half-plane ℋ. It follows that N1 = NΩ(Γ1) is discrete; here N1is the normalizer of Γ in Ω, the group of (conformal) automorphisms of ℋ. An automorphism of S can be lifted to a coset of Nl/Γl. Hence C(S), the group of automorphisms of S, is isomorphic to Nl/Γ1. The order of C = C(S) equals the index of Γ1 in N1, which in turn equals ⃒Γ1⃒ / ⃒Nl⃒, where ⃒Nl⃒ is the hyperbolic area of a fundamental region of Nl. Since Γ1 uniformizes a surface, we have ⃒Γ1⃒ = 4π(g – 1), while, by Siegel's results [7], ⃒N1 ⃒ ≧ π/21 and N1 can only be the triangle group (2, 3, 7). Hence in all cases the order of C(S) is at most 84(g–1), an old result of Hurwitz [1]. The surfaces that permit a maximal automorphism group (= automorphism group of maximum order) can therefore be obtained by studying the finite factor groups of (2, 3, 7). Such a treatment, purely algebraic in nature, has been promised by Macbeath [5].

The object of this paper is to consider two easy propositions concerning bounded approximate identities and show that they do not extend to unbounded approximate identities. The propositions are as follows.

Proposition 1.1. Every bounded left approximate identity in a normed algebra is a left approximate identity for the completion.

Proposition 1.2. Every bounded left approximate identity in a separable normed algebra has a subsequence which is a left approximate identity.

In his book Transversal Theory [3], L. Mirsky has remarked that “At present, the relation between Dilworth's decomposition theorem…and transversal theory is rather tenuous; but further study may reveal unexpected connections”. Some of these connections can perhaps now be seen a little more clearly; and our purpose in this note is to make one or two observations in this regard. Throughout, all sets considered are finite.

Let L = L( +, v, ＾) be a lattice-ordered group, or l-group (Birkhoff [1, p. 214]). Two elements a and b of L will be called disjoint if a > 0, b > 0, and a ＾; b = 0. It is easily seen that if L does not contain two disjoint elements, then it is linearly ordered (and, of course, conversely). What can we say about Z-groups containing two but not more than two mutually disjoint elements?

Let Aand B be linearly ordered groups (o-groups), and let A ⋏ B be the cardinal sum of A and B. That is, A ⋏ B is the direct sum of A and B, and (a, b) is positive in A + B if and only if a is positive in A and b is positive in B. An l-group L containing A ⋏ B as a convex normal subgroup (or Z-ideal) is called a lexico-extension of A ⋏ B if every positive element of L not in A ⋏ B exceeds every element of A ⋏ B. It then follows (subsection 2.9 below) that L/(A ⋏ B) is an o-group. Such an l-group L is easily seen to satisfy the following condition: (D)

There exists a pair of disjoint elements in L, but no triple of pairwise disjoint elements exists in L.

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7].

All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if

(ρ): for every ring R and every left (right) ideal I of R it follows S(I)⊆S(R).

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

Walker [11] describes a new class of translation planes W(q) of order q2, q ≡ 5(mod 6), with kernel GF(q). A plane in this class has several interesting properties, but we shall be only interested in the following ones possessed by its collineation group G: (i) G is transitive on the affine points of W(q), and (ii) G fixes a point the line at infinity of W(q), and is transitive on the other points of l∞ The smallest member of this class, W(5), also satisfies: (iii) for an affine point the subgroup G is transitive on the affine points ≠ of the line Note also that since W(5) is a translation plane, replacing G with G in (ii) we get a fourth property, call it (iv), satisfied by G. (See Lemma 2.)

The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [5, 6]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [10] and R. K. Dennis [7]. The tensor product of crossed complexes was introduced by R. Brown and the second author [3] in connection with the fundamental crossed complex π(X) of a filtered space X, which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π(X ⊗ Y) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

In this note a characterization of subnormality of operators on Hilbert space is given. The characterization is in terms of a sequence of polynomials in the operator and its adjoint reminiscent of the binomial expansion in commutative algebras. As such no external Hilbert spaces are needed, nor is it necessary to introduce forms dependent on arbitrary sequences of vectors from the Hilbert space.