The nth order polylogarithm Lin(z) is defined for |z| ≦ 1 by

([4, p. 169], cf. [2, §1. 11 (14) and § 1. 11. 1]). The definition can be extended to all values of zin the z-plane cut along the real axis from 1 to ∝ by the formula

[2, §1. 11(3)]. Then Lin(z) is regular in the cut plane, and there is a differential recurrence relation [4, p. 169]

It is convenient to extend the sequence Lin(z) backwards in the manner suggested by (2) and define

Then Li1(z)= – log(l–z), and Lin(z) is a rational function of z for n= 0, – 1, – 2,…. Formula (2) now holds for all integers n.

Let G, H, K be groups such that G is normal in K and G ⊆ H ⊆ K. Let I(H, K) be the set of inner automorphisms of K restricted to H; thus α ∊ I(H, K) if and only if, for some κ∊ K, α(h) = k-1hk for all h ∊ H. Let φ be an isomorphism of H/G onto a subgroup Hφ/G of K/G. An isomorphism Φ of H onto H(φ) is called an extension of ø if

Φ(h)G = φ(hG) for all h∊H.

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

1. It is a well known theorem due to Frobenius that the number of solutions of the equation

Xn = 1

in a finite group G, is a multiple of the greatest common divisor (n, g) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equation

Xn = a

where a is a fixed element of G, is a multiple of (n, ga), ga being the order of the centralizer Z(a) of ain G.

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

Given a probability space (X, ℱ, μ) and a σ-algebra A ⊂ ℱ, arguably the most powerful tool in gaining information about an ℱ-measurable function f from restricted knowledge of -measurability is that of the conditional expectation E(f | ); written throughout the remainder of this note. Two properties of conditional expectation that may be exploited to gain information, but which also limit conditional expectation's use are the following.

On a semigroup S let the relation ℛ*, sometimes denoted by ℛ, be defined by xℛ*y[(sx = txsy = ty]. A semigroup S is called left type-A, iff the set Es of idempotents of S forms a semilattice under multiplication, each element x of Sis ℛ* related to a (necessarily unique) idempotent x+, and xe = (xe)+x for all x ∈ S, е ∈ Es.

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the set

If V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

The object of this paper is to evaluate a few infinite integrals involving E-functions by applying the Parseval-Goldstein [1] theorem of Operational Calculus; that, if

and

then

when the integrals are convergent.

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].